On the Role of Charmed Meson Loops in Charmonium Decays
aa r X i v : . [ h e p - ph ] N ov INT-PUB-11-055
On the Role of Charmed Meson Loops in Charmonium Decays
Thomas Mehen and Di-Lun Yang Department of Physics, Duke University, Durham, NC 27708, USA (Dated: June 18, 2018)
Abstract
We investigate the effect of intermediate charmed meson loops on the M1 radiative decays
J/ψ → η c γ and ψ ′ → η ( ′ ) c γ as well as the isospin violating hadronic decays ψ ′ → J/ψ π ( η ) using heavyhadron chiral perturbation theory (HH χ PT). The calculations include tree level as well as one loopdiagrams and are compared to the latest data from CLEO and BES-III. Our fit constrains thecouplings of 1S and 2S charmonium multiplets to charmed mesons, denoted g and g ′ , respectively.We find that there are two sets of solutions for g and g ′ . One set, which agrees with previousvalues of the product g g ′ extracted from analyses that consider only loop contributions to ψ ′ → J/ψ π ( η ), can only fit data on radiative decays with fine-tuned cancellations between tree leveldiagrams and loops in that process. The other solution for g and g ′ leads to couplings that aresmaller by a factor of 2.3. In this case tree level and loop contributions are of comparable sizeand the numerical values of the tree level contributions to radiative decays are consistent withestimates based on the quark model as well as non-relativistic QCD (NRQCD). This result showsthat tree level HH χ PT couplings are as important as the one loop graphs with charmed mesons inthese charmonium decays. The couplings g and g ′ are also important for the calculations of thedecays of charmed meson bound states, such as the X (3872), to conventional charmonia. α s and v c , where v c is the relative velocity of thecharm-anticharm quarks. Despite many successes there remain specific transitions that arenot well understood quantitatively. Examples of decays that are not completely understoodare the hadronic decays ψ ′ → J/ψ ( π , η ), and the radiative decays to J/ψ → η c γ and ψ ′ → η ( ′ ) c γ . The hadronic decays violate isospin, in the case of a final state with π , or SU (3), when the final state is η . As a consequence the ratio of this decay is sensitive to lightquark masses [3, 4]. The value of the light quark mass ratio extracted from the measureddecay rates [5, 6], m u /m d = 0 . ± .
01, differs significantly from the result extracted frommeson masses in chiral perturbation theory, m u /m d = 0 .
56 [7, 8]. For the radiative decaysthe experimentally measured rates differ from quark model expectations. For example, anon-relativistic quark model calculation of
J/ψ → η c γ ( ψ ′ → η c γ ) yields a prediction of ≈ ≈
0) keV, whereas the experimental results are 1.57 ± ± J/ψ → η c γ , and the authors show that O ( v ) corrections can lower the rate so that the theoretical prediction is consistent with data.However, no attempt has been made to understand radiative decays of ψ ′ in this framework.For reviews of these puzzles and others in charmonium physics, see Refs. [9, 12, 13].Recently, Ref. [14] proposed that the hadronic decays mentioned above are dominatedby loop diagrams with virtual D mesons. The decays are calculated using Heavy HadronChiral Perturbation Theory (HH χ PT) [15–17], in which the charmonia are treated non-relativistically and coupled to the D mesons and Goldstone bosons in a manner consistentwith heavy quark and chiral symmetries. In a non-relativistic theory the D meson kineticenergy scales as m D v and momentum scales as m D v , where m D is a D meson mass and v ≈ / D mesons in the loops. With this scaling, Ref. [14]showed that the loop diagrams with D mesons should be enhanced over tree level couplings The decay rate ψ ′ → η c γ vanishes in the non-relativistic quark model due to the vanishing overlap of theorbital wavefunctions of the ψ ′ and the η c , and is no longer zero once relativistic corrections are takeninto account. However, quark models that include relativistic corrections still have trouble reproducingthe correct rate for ψ ′ → ηγ [9].
2y a factor of 1 /v . The rates for ψ ′ → J/ψπ and ψ ′ → J/ψη are sensitive to the product g g ′ , where the J/ψ coupling to D mesons is g and the ψ ′ coupling to D mesons is g ′ .Ref. [14] found a value of g g ′ consistent within errors with the two experimentally measuredrates. This resolves the disagreement between the value of m u /m d extracted from thesedecays and other extractions, since the prediction for the ratio of rates in terms of m u /m d relied on the rates being dominated by the tree level HH χ PT coupling. The value of g g ′ extracted by Ref. [14] is consistent with power counting estimates of g and g ′ , which areboth expected to be ∼ ( m c v c ) − / up to constants of order unity. Other hadronic andradiative charmonium decays are also analyzed within the same formalism in Refs. [18–20].The goal of this paper is to apply the same theory to the radiative decays mentionedabove. One of our aims is to check whether the theory can also successfully resolve puzzlesin radiative decays as one would hope. It is also important to check that couplings extractedfrom the hadronic decays are consistent with data on radiative decays. An important aspectof our analysis is that unlike Refs. [14, 18], tree level counterterms are included in ourcalculations of both hadronic and radiative decays. Ref. [18] argued for an additional factorin the loop graphs of 1 / (4 πv c ) ≈ . − .
6, for v c ≈ . − .
3, which would compensate the1 /v enhancement of the loops. This factor, and the fact that v is not very small, supportincluding both the loops and tree level interactions in the calculation, which we will do inthis paper. This can have an important impact of the extracted values of the couplings g and g ′ . Finally, an additional motivation for our analysis is that the extracted couplings areimportant for the physics of the X (3872) and other recently discovered charmonium boundstates that have been interpreted as charmed meson molecules. If the X (3872) is a charmedmeson bound state, then the coupling g ( g ′ ) is an important theoretical input for calculationsof X (3872) → J/ψ ( ψ ′ )+ X , so extraction of g and g ′ is relevant to unconventional as well asconventional charmonia. For theoretical calculations of X (3872) to conventional charmoniausing effective field theory, see Refs. [21–23].Our main result is that in order to obtain a consistent fit to both radiative decays aswell as the hadronic decays considered in Refs. [14, 18], counterterm contributions mustbe included and the values of g and g ′ will then be smaller than estimated in an analysiscontaining only the loop diagrams by a factor of 2.3. This decreases the overall size ofthe loop amplitude by a factor of 5. It is not possible to get reasonable agreement withradiative decay data without including counterterms. Since NRQCD is the microscopic3heory of charmonia, and does not include loop effects from charmonia, one is tempted toidentify the result of a calculation of the J/ψ → η c γ amplitude in NRQCD with the treelevel coupling in HH χ PT. This is somewhat tenuous as the bare coupling in our theoryhas an infinite piece that must cancel the linear divergence in the meson loop integrals. Nevertheless, we regard it as satisfying that the size of the counterterms we extract in ourfit with the smaller values of g and g ′ are consistent within a factor of 2 with the quarkmodel and NRQCD calculations of the radiative transitions. For other extractions of thecouplings g and g ′ in different theoretical frameworks, see, e.g., Refs. [24–30]. In Refs.[31, 32], the charmed meson loop corrections to radiative J/ψ and ψ ′ decays are studied ina version of HH χ PT with relativistic propagators and couplings, as well as form factors atthe vertices that regulate ultraviolet divergences. The form factors introduce an additionalparameter into the calculations. These authors did not attempt to simultaneously fit thehadronic decays but used values of g and g ′ consistent with those obtained in Refs. [14, 18].Their results are also consistent with the experimental data on the radiative decays.The effective HH χ PT Lagrangian relevant to the hadronic decays is [18, 21, 33] L = T r [ H † a (cid:18) i∂ + ∇ m D (cid:19) H a ] + ∆4 T r [ H † a ~σH a ~σ ] − g T r [ H † a H b ~σ · ~u ab ] (1)+ i A (cid:0) T r [ J ′ σ i J † ] − T r [ J † σ i J ′ ] (cid:1) ∂ i ( χ − ) aa + i g T r [ J † H a ~σ · ←→ ∂ ¯ H a ] + H.c. .
Here H a = V a · ~σ + P a and ¯ H a = − ¯ V a · ~σ + ¯ P a are the charmed and anti-charmed mesonmultiplets with V a and P a denoting the vector and pseudoscalar charmed mesons, respec-tively, and J ( ′ ) = ~ψ ( ′ ) · ~σ + η ( ′ ) c denotes the charmonium multiplets with ~ψ ( ′ ) and η ( ′ ) c . The ~σ are the Pauli matrices, a and b denote flavor indices, and A ←→ ∂ B = A ( ~∂B ) − ( ~∂A ) B . Thefirst two terms in Eq. (1) are kinetic terms for the charmed mesons, ∆ = m D ∗ − m D is thehyperfine splitting, and m D ( m D ∗ ) is the mass of pseudoscalar (vector) charmed meson. Thethird term contains the interactions of D mesons with the Goldstone boson fields which arecontained in u = exp ( iφ/ √ F ) where φ is a 3 × F = 92 . D mesonswhich are not explicitly shown. The tree level couplings for ψ ′ → J/ψπ ( η ) come from theterm with coupling constant A . The factor χ − is defined by χ − = u † χu † − uχ † u , where χ = 2 B · diag ( m u , m d , m s ), m u m d and m s are the light quark masses and B = |h | ¯ qq | i| . This linear divergence is absent in dimensional regularization. D mesons via the last term with coupling g . Thesame term, with J and g replaced with J ′ and g ′ , couples the 2 S charmonia to charmedmesons.The tree level decay amplitudes are [18], i M ( ψ ′ → J/ψπ ) = i Aǫ ijk q i ǫ ψ ′ j ǫ J/ψk B du i M ( ψ ′ → J/ψη ) = i (8 / √ Aǫ ijk q i ǫ ψ ′ j ǫ J/ψk B sl , (2)where B du = B F ( m d − m u ) and B sl = B F ( m s − m u + m d ). To leading order in the chiralexpansion, these factors may be expressed in terms of light meson masses: B du = ( m K − m K + + m π + − m π ) /F and B sl = (3 / m η − m π ) /F . The π − η mixing must also beincluded, and the mixing angle is ǫ π η = 1 √ m K − m K + + m π + − m π m η − m π . (3)When this mixing is included the first matrix element in Eq (2) is multiplied by 3/2. Theloop diagrams contributing to the decay have been evaluated in Refs. [14, 18]. Since thedecay to π ( η ) vanishes in the isospin ( SU (3)) limit, the diagrams cancel in the sum over D , D + , and D + s appearing in the loop in the limit that all these mesons are degenerate.Mass differences between the mesons render the cancellation incomplete and are responsiblefor the finite contribution.For electromagnetic decays, we need to add couplings to the magnetic field and gaugethe interactions in Eq. (1). The tree level coupling of the charmonia to the magnetic fieldsis given by [34, 35] ρ T r [ J ~B · ~σJ † ] + ρ ′ T r [ J ′ ~B · ~σJ † ] + H.c. ) + ρ ′′ T r [ J ′ ~B · ~σJ ′ † ] , (4)where ~B is the magnetic field. Due to the presence of Pauli matrices these terms break heavyquark spin symmetry. The first term is responsible for the decay J/ψ → η c γ , the second for ψ ′ → η c γ , and the third for ψ ′ → η ′ c γ . For the loop corrections to the radiative decays, wemust also include the coupling of the charmed mesons to the magnetic field, which is givenby [33, 36] eβ T r [ H † a H b ~σ · ~BQ ab ] + e m c Q ′ T r [ H † a ~σ · ~BH a ] , (5)where Q ab = diag (2 / , − / , − / Q ′ = 2 /
3, and m c is the mass of charm quark. Theseterms are responsible for the decays D ∗ → Dγ . Including leading as well as Λ QCD /m c /ψJ/ψ J/ψJ/ψD ¯ D D ∗ γη c D ¯ D ∗ D ∗ γη c D ∗ ¯ D ∗ D γη c D ∗ ¯ D D ∗ γη c ( a ) ( b )( c ) ( d ) J/ψ D ∗ ( e ) ¯ D ∗ D ∗ γη c FIG. 1: Triangle diagrams with intermediate charmed meson loops. The charmed meson couplingsto photon come from Eq. (5).
J/ψ γη c D ∗ D ∗ ¯ D D γη c ¯ D ∗ J/ψ D
FIG. 2: Triangle diagrams with intermediate charmed meson loops. The charmed meson couplingsto photon come from gauging the kinetic terms in Eq. (1). suppressed terms is crucial for reproducing observed D ∗ → Dγ rates [36]. Ref. [33] findsthat a good fit to the experimental rates is obtained for the values m c = 1 . β = 3 . − .These couplings enter the radiative decays of charmonia through the triangle diagramsshown in Fig. 1. There are also interactions that arise from gauging the derivatives inEq. (1). Gauging the derivatives in the kinetic term for the D mesons leads to couplingsto the photon which contribute to the radiative decays via triangle loop diagrams shownin Fig. 2. Gauging the coupling g ( ′ )2 leads to a contact interaction that directly couplescharmonia, heavy mesons and the photon field, which is given by − eg T r [ J † H a ~σ · ~A ¯ H a ] + H.c. (6)6 /ψ η c D ¯ D ∗ γ ( a ) J/ψ ( b ) γ D ∗ ¯ D η c J/ψ γ ( c ) D ¯ D ∗ η c J/ψ η c γD ∗ ¯ D ( d ) FIG. 3: Contact diagrams with intermediate charmed meson loops, where a = 2 or 3 only, i.e., only charged and strange D mesons appear in the interactionterm of Eq. (6). The loop diagrams with contact interactions are shown in Fig. 3.The tree level amplitude for the J/ψ → η c γ decay, for example, is i M = ρ ǫ ijk q i ǫ γj ǫ J/ψk , (7)where q denotes the momentum of photon, and ǫ γ and ǫ J/ψ are the polarization vectors ofthe photon and
J/ψ , respectively. The corresponding decay rate isΓ[
J/ψ → η c γ ] = 18 π |√ m J/ψ m η c M | | ~q | m J/ψ , = ρ π m η c m J/ψ | ~q | . (8)Here the factor √ m J/ψ m η c comes from the normalization of nonrelatvistic fields in HH χ PT.In the non-relativistic quark model, ρ = 2 ee c /m c ≈ . − , and the predicted decay rateis about a factor of 2 too large. The contribution to the amplitude from meson loops is alsoproportional ǫ ijk q i ǫ γj ǫ J/ψk so the loops give an additive shift to the ρ term for each decay. Weevaluate the loops in pure dimensional regularization so linear divergences do not appearand the corrections from all loops are finite. The full rate is Eq. (8) with M replaced by M full , where the M full includes both the tree level interaction and the contributions fromneutral, charged, and strange meson loops. The explicit expression for M full can be foundin the Appendix.Before proceeding to our fits to the data, we will briefly discuss the power counting forthe diagrams we have shown. As stated earlier, for non-relativistic D mesons one takes7 ∼ m D v , p ∼ m D v , so the propagators scale as ( m D v ) − and the loop integrationmeasure is m D v . The vertices coupling the charmonium to D mesons carry a factor of p ∼ m D v . To estimate v one may take the difference between an external and two internalmesons, so m D v = | m charmonium − m D pair | where m charmonium is the mass of one of theexternal charmonia and m D pair is the mass of two D mesons in the loop. This leads toan estimate ranging from v = 0 .
09 ( m charmonium = m ψ ′ and m D pair = m D + m D ∗ ) to v = 0 . m charmonium = m J/ψ and m D pair = 2 m D ∗ ). Naively with this counting the trianglediagrams scale as ( m D v )( m D v ) − ( m D v ) q = m D vq . The first factor comes from the loopintegration factor, the second from the propagators, the third factor from the derivativecouplings of charmonium to D mesons, and the factor of q is the photon or pion momentumwhich comes from the coupling of these particles to D mesons. The diagrams with thecontact interaction scale as ( m D v )( m D v ) − q = m D vq which is the same as the trianglegraph. This is because there is no derivative in the contact interaction with the photon, thederivative in the charmonium D meson coupling must turn into the factor of q required bygauge invariance or chiral symmetry and there are only two propagators. The factor q iscommon to all diagrams including the tree level diagrams. Factors of m D are compensatedby other dimensionful couplings so we will focus only on counting powers of v from here on.So the triangle graphs and graphs with the contact interaction are v suppressed relative tothe tree level interactions. However, these hadronic decays violate either isospin or SU (3)and the radiative decays violate heavy quark symmetry so there are cancellations betweengraphs due to heavy meson mass differences that are missed by this power counting. One canformally modify the power counting counting in the following way. The inverse propagatorfor a non-relativistic meson can be written as E − p m D + b + δ , where the residual mass termin the propagator has been split into a term b which is common to all D meson states anda term δ contains SU (3) breaking and hyperfine splittings that are different for different D mesons. Expand the D meson propagators as1 E − p m D + b + δ = 1 E − p m D + b − δ (cid:16) E − p m D + b (cid:17) + ... . The graph in which all propagators contribute only the first term is zero by symmetry.In order to get a non-vanishing result at least one propagator in the graph must give acontribution from the second term, then the power counting says the graph is enhanced bya factor of δ/m D v , which makes the graph 1 /v enhanced rather than v suppressed relative8o the tree level diagrams [14, 18]. Since v is not very small this could be compensatedby other numerical factors. In practice it is easier to simply calculate the graphs with theunexpanded propagators but expanding the propagator makes it clear that after summingover all graphs one gets a 1 /v enhancement. In this paper, we will take the viewpoint thatthe leading one loop diagrams are of roughly the same size as the tree level contributionsand include both in the decays, then try to simultaneously fit the radiative and hadronicdecays mentioned above.A separate question is whether higher order chiral corrections are under control. Certainlysome chiral corrections are suppressed as argued for different charmonium radiative decaysin Ref. [20]. But in a subgraph with a ladders of single pion exchanges between a pair of D mesons, non-relativistic power counting shows that the ladder with n + 1 single pionexchanges is suppressed relative to one with n pion exchanges by a factor g m D p/ (8 πF ) = p/ (320 M eV ) [37] where p is the relative momentum of the D mesons. This would require p = m D v with v < .
08 to be less than 1 /
2. In some channels a resummation of single pionexchanges may be needed to do accurate calculations. Such a resummation is beyond thescope of this paper. Here we are simply interested in the impact that including the tree levelinteractions and simultaneously fitting the radiative and hadronic decays has on the valuesof g and g ′ and therefore the size of D meson loop contributions to charmonium decays.To constrain the parameters g and g ′ , we determine the parameter A and the product g g ′ from the measured rates for Γ[ ψ ′ → J/ψπ ] and Γ[ ψ ′ → J/ψη ]. Because the predictionsfor the decay rates are quadratic in g g ′ , this does not completely determine g g ′ , but yieldstwo possible solutions. Then we fix the relative size of the two couplings using the relation g = g ′ q m ′ ψ /m J/ψ , which follows if the dimensionless coupling of the
J/ψ and ψ ′ to D mesons is the same [14, 18]. Once g and g ′ are determined this way from the hadronicdecays, the only parameters remaining in the radiative decays are ρ , ρ ′ , and ρ ′′ , which canbe determined from the three decay rates Γ[ J/ψ → η c γ ], Γ[ ψ ′ → η c γ ], and Γ[ ψ ′ → η ′ c γ ]. Theresults of determining A , g , and g ′ are shown in the first three columns of Table 1. Onepossible fit to the hadronic decays yields A = − .
36 10 − GeV − and g g ′ = 1 .
73 GeV − .This is a very small value of A , almost two orders of magnitude smaller than the estimate A ∼ / (2 m c ) in Ref. [18]. This fit yields a value of g g ′ similar to that of Refs. [14, 18]. In this case, we get a value of g g ′ that is a factor of two smaller than Refs. [14, 18] because our calculations ( GeV − ) g ( GeV − / ) g ′ ( GeV − / ) ρ ( GeV − ) ρ ′′ ( GeV − ) ρ ′ ( GeV − ) − . +4 × − − × − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . − . +4 × − − × − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . .
275 0 .
263 0 . ρ , ρ ′ and ρ ′′ are shown in the bottom line. Errors are due to experimental uncertainties only. The second possible fit is A = 2 .
57 10 − GeV − and g g ′ = 0 .
329 GeV − , a value 5.3 timessmaller than the first fit. The value of A is closer to the estimate of 1 / (2 m c ), but still afactor of 10 smaller. The results of fitting the parameters ρ , ρ ′ , and ρ ′′ , are shown in the lastthree columns of Table 1. For each choice of A , g , and g ′ , there are two possible solutionsfor ρ , ρ ′ , or ρ ′′ , for a total of four possible solutions. The values of ρ and ρ ′′ are much closerto the quark model predictions (shown in the bottom row of Table 1) in the fit with a smallervalue of g g ′ . The extracted value of ρ ′ does not come close to the quark model prediction ofRef. [9], but this model does not give a good prediction for the rate ψ ′ → η c γ . In the fits withthe larger value of g g ′ the extracted values of ρ , ρ ′ , and ρ ′′ are much larger. This indicatesthat for these choices of parameters fine tuned cancellations between the tree level and loopdiagrams are required to fit the data. This can also be clearly seen in Table II, where wegive the loop contribution to the decay for each fit. For the first fit with g g ′ = 1 .
73 GeV − ,Γ[ J/ψ → η c γ ] is over predicted by a factor of 100 and Γ[ ψ ′ → η ′ c γ ] is over predicted by afactor of 20 without the counterterm contribution. Thus, in order to fit these decays, fine-tuned cancellations between loop and tree level contributions must occur. Though the loopcontributions by themselves do not do a good job of producing the radiative decay rates forthe smaller value of g g ′ , the discrepancy is not nearly as large. of the loop amplitudes for ψ ′ → J/ψπ ( η ) disagree with the analytic results of Refs. [14, 18] by an overallfactor of two. This is because we include graphs in which the π or η couples to the ¯ D ( ∗ ) mesons, insteadof the D ( ∗ ) mesons, that are omitted in Refs. [14, 18]. it 1 Fit 2 QM PDG, BES III, and CLEOΓ[ J/ψ → η c γ ] loop +10 − keV 5 . +0 . − . keV 2.9 keV 1.58 ± ψ ′ → η ′ c γ ] loop . +0 . − . keV 0 . +0 . − . keV 0.21 keV 0.143 ± ± ψ ′ → η c γ ] loop . +0 . − . MeV 597 +97 − keV 9.7 keV 0.97 ± g g ′ iscompared with the results in the quark model [9] and experimental data. Fit 1 corresponds to g g ′ = 1 . +0 . − . (GeV − ), Fit 2 corresponds to g g ′ = 0 . +0 . − . (GeV − ). For the decay ψ ′ → η c γ there are rather severely fine tuned cancellations between loopdiagrams and tree level contributions for both fits. The photon energies in the decays J/ψ → η c γ , ψ ′ → η c γ , and ψ ′ → η ′ c γ are 114 MeV, 638 MeV, and 49 MeV, respectively. Thephoton energy in the second decay may be too large for either the quark model or low energyeffective theory to be accurate. As an alternative approach, one can simply try to extract g and g ′ independently from the radiative decays J/ψ → η c γ and ψ ′ → η ′ c γ , using the quarkmodel [9] to estimate the parameters ρ and ρ ′′ . Since the decay J/ψ → η c γ is quadraticin g and the decay ψ ′ → η ′ c γ is quadratic in g ′ , there are two possible solutions for eachparameter. We find g = 0 . +0 . − . GeV − / or 0 . +0 . − . GeV − / and g ′ = 0 . +0 . − . GeV − / or 0 . +0 . − . GeV − / . Note that the value of g g ′ obtained this way is also smallerthan the value obtained in the first fit to the combined hadronic and radiative decays. Alsothe ratio g ′ /g obtained using the the smaller two central values is 1 .
04 while using thelarger two central values the ratio is 1 .
3. Both of these are a little larger than one expectsfrom the hypothesis g ′ /g = p m J/ψ /m ψ ′ = 0 . J/ψ → η c γ and ψ ′ → η ( ′ ) c γ includingtree level and one loop diagrams with charmed mesons in HH χ PT. We combined our resultswith the decay rates for ψ ′ → J/ψπ ( η ) found in Refs. [14, 18], used the relationship g ′ = g p m J/ψ /m ψ ′ , and fit the five remaining coupling constants simultaneously. Includingtree level couplings is essential for simultaneously reproducing all the decay rates. A smallervalue of g g ′ = 0 .
33 GeV − is required to avoid large cancellations between tree level andcharmed meson loop contributions to the radiative decay. The tree level couplings ρ and ρ ′′ in this fit are consistent (to within a factor of 2) with expectations based on the quarkmodel and NRQCD. 11 cknowledgments This work was supported in part by the Director, Office of Science, Office of NuclearPhysics, of the U.S. Department of Energy under grant numbers DE-FG02-05ER41368(T.M.) and DE-FG02-05ER41367 (D.Y.). We thank the Department of Energy’s Institutefor Nuclear Theory at the University of Washington for its hospitality during the completionof this work.
APPENDIX
In this Appendix, we calculate the loop diagrams for the M1 radiative decays. Wepresent the calculation of
J/ψ → η c γ , the calculations of the decays ψ ′ → η c γ ( ψ ′ → η ′ c γ )are obtained by replacing m J/ψ with m ψ ′ and g with g g ′ ( g ′ ).The triangle loop diagrams in Fig. 1 have a similar form as the triangle loop diagrams inthe hadronic decays, therefore the notation used here will be the almost the same as thatof Ref. [18]. We refer the reader to that paper for explicit expressions for the integrals. Theamplitude from Fig. 1(a) is i M a = − ig λ Z d l (2 π ) ǫ ijk q i ǫ γj (2 l − q ) k ~ǫ J/ψ · ~l l − ~l m D + iǫ )( l + ~l m D + b DD − iǫ )( l − q − ( ~l − ~q ) m D ∗ − ∆ + iǫ )= 4 g λ ǫ ijk q i ǫ γj ǫ J/ψk | ~q | I (2)1 ( q, m D , m D , m D ∗ ) , (9)where b DD = 2 m D − m J/ψ , λ = ( eβ + em c ) is relevant for loops with neutral D mesonsand λ = − ( eβ − em c ) is relevant for loops with charged and strange D mesons. Here I (2)1 ( q, m , m , m ) only differs from the function defined in Ref. [18] by omitting a factor of m m m from the denominator. Fig. 1(b) contributes i M b = 2 g λ ǫ ijk q i ǫ γj ǫ J/ψk | ~q | (10) × (2 I (2)0 ( q, m D , m D ∗ , m D ∗ ) + 4 I (2)1 ( q, m D , m D ∗ , m D ∗ ) − I (1) ( q, m D , m D ∗ , m D ∗ )) , where the functions I (2)0 ( q, m , m , m ) and I ( q, m , m , m ) are again the same as functionsin Ref. [18] up to a factor of m m m . Fig. 1(c) contributes i M c = 2 g λ ǫ ijk q i ǫ γj ǫ J/ψk | ~q | (11) × (2 I (2)0 ( q, m D ∗ , m D ∗ , m D ) + 6 I (2)1 ( q, m D ∗ , m D ∗ , m D ) − I (1) ( q, m D ∗ , m D ∗ , m D )) . i M d = 2 g λ ǫ ijk q i ǫ γj ǫ J/ψk | ~q | (12) × (2 I (2)0 ( q, m D ∗ , m D , m D ∗ ) + 4 I (2)1 ( q, m D ∗ , m D , m D ∗ ) − I (1) ( q, m D ∗ , m D , m D ∗ )) , where λ = − ( eβ − em c ) is relevant for loops with neutral D mesons and λ = ( eβ + em c )is relevant for loops with charged and strange D mesons. Finally, Fig. 1(e) gives i M e = 2 g λ ǫ ijk q i ǫ γj ǫ J/ψk | ~q | (13) × (2 I (2)0 ( q, m D ∗ , m D ∗ , m D ∗ ) + 8 I (2)1 ( q, m D ∗ , m D ∗ , m D ∗ ) − I (1) ( q, m D ∗ , m D ∗ , m D ∗ )) . In addition to the triangle diagrams with the couplings of D and D ∗ mesons to the magneticfield from Eq. (5), there are also two triangle diagrams with the coupling of the photon tocharged D and D ∗ mesons that arises due to gauging their kinetic terms. These are shownin Fig. 2. The sum of these two diagrams yields i M = 4 g eǫ ijk q i ǫ γj ǫ J/ψk | ~q | (cid:18) m D I (2)1 ( m D , m D ∗ , m D ) − m D ∗ I (2)1 ( m D ∗ , m D , m D ∗ ) (cid:19) . (14)So far we have only included the interactions coupling the photon to D and D ∗ mesons.There are additional diagrams where the photons couple to ¯ D and ¯ D ∗ mesons that give anequal contribution.Fig. 3 shows the loop diagrams with the contact interaction that arises from gauging thecoupling g . Fig. 3(a) yields i M a = i g e Z d l (2 π ) ǫ ijk (2 l + q ) i ǫ γj ǫ J/ψk l − ~l m D + iǫ )( l + q + ( ~l + ~q ) m D ∗ + b DD ∗ − iǫ )= − g eǫ ijk q i ǫ γj ǫ J/ψk I ′ ( m D , m D ∗ ) , (15)where I ′ ( m D , m D ∗ ) = µ DD ∗ π (cid:18) m D ∗ − m D m D ∗ + m D (cid:19) r µ DD ∗ m D + m D ∗ | ~q | + 2 µ DD ∗ ( b DD ∗ + q ) , (16)for µ DD ∗ = m D m D ∗ / ( m D + m D ∗ ) and b DD ∗ = m D + m D ∗ − m J/ψ . Fig. 3(b) is related toFig. 3(a) by charge conjugation so i M b = i M a = − g eǫ ijk q i ǫ γj ǫ J/ψk I ′ ( m D , m D ∗ ) . (17)13he graphs in Fig. 3(c) and Fig. 3(d) both vanish, so the total contribution to the amplitudefrom loops with contact interactions is 2 i M a . Only diagrams with charged and strange D mesons in the loop will contribute.The total amplitude from loop diagrams in Fig. 1 with neutral D mesons is given by i M n = ǫ ijk ~q i ǫ γj ǫ J/ψk { g | ~q | [4 λ I (2)1 ( q, m D , m D , m D ∗ ) + 2 λ (2 I (2)0 ( q, m D , m D ∗ , m D ∗ )+4 I (2)1 ( q, m D , m D ∗ , m D ∗ ) − I (1) ( q, m D , m D ∗ , m D ∗ )) + 2 λ (2 I (2)0 ( q, m D ∗ , m D ∗ , m D )+6 I (2)1 ( q, m D ∗ , m D ∗ , m D ) − I (1) ( q, m D ∗ , m D ∗ , m D )) + 2 λ (2 I (2)0 ( q, m D ∗ , m D , m D ∗ )+4 I (2)1 ( q, m D ∗ , m D , m D ∗ ) − I (1) ( q, m D ∗ , m D , m D ∗ )) + 2 λ (2 I (2)0 ( q, m D ∗ , m D ∗ , m D ∗ )+8 I (2)1 ( q, m D ∗ , m D ∗ , m D ∗ ) − I (1) ( q, m D ∗ , m D ∗ , m D ∗ ))] } . (18)Adding the five diagrams with the photon coupling to a ¯ D or ¯ D ∗ doubles this contribution.The contribution from diagrams of Fig. 1 with charged and strange charmed mesons in theloops is obtained by substituting λ with λ and λ with λ . 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