Electromagnetic Interactions in

arXiv:hep-ph/9209239v1 15 Sep 1992HUTP-92/A043Electromagnetic Interactions inHeavy Hadron Chiral TheoryPeter Cho † and Howard GeorgiLyman Laboratory of PhysicsHarvard UniversityCambridge, MA 02138Electromagnetic interactions are incorporated into Heavy Hadron Chiral PerturbationTheory. Short and long distance magnetic moment contributions to the chiral Lagrangianare identified, and M1 radiative decays of heavy vector mesons and sextet baryons arestudied.

Using recent CLEO D∗branching fraction ratio data, we fit the meson couplingto the axial vector Goldstone current and find g21 = 0.34±0.48 for mc = 1700MeV. Finally,we obtain model independent predictions for total and partial widths of charm and bottomvector mesons.9/92† Address after Sept 21, 1992: California Institute of Technology, Pasadena, CA 91125.

A synthesis of Chiral Perturbation Theory and the Heavy Quark Effective Theory(HQET) has recently been developed [1–5]. This hybrid effective theory describes lowenergy strong interactions between light Goldstone bosons and hadrons containing a singleheavy quark.

Weak b →c transitions among heavy meson or baryon states can also beincorporated into this framework.In this letter, we extend the theory’s formalism toinclude electromagnetism and then study the radiative decays of heavy vector mesons andsextet baryons.To begin, we briefly review the basic elements of Heavy Hadron Chiral PerturbationTheory (HHCPT). 1 The Goldstone bosons resulting from the chiral symmetry breakdownSU(3)L × SU(3)R →SU(3)L+R appear in the pion octetπ =8Xa=1πaT a =1√2q12π0 +q16ηπ+K+π−−q12π0 +q16ηK0K−K0−q23η(1)and are associated with the pion decay constant f ≈93 MeV.

These fields are arrangedinto the exponentiated matrix functions Σ = e2iπ/f and ξ = “√Σ” = eiπ/f that transformunder the chiral symmetry group asΣ →LΣR†ξ →LξU † = UξR†. (2)Here L and R represent global elements of SU(3)L and SU(3)R, while U acts like a lo-cal SU(3)L+R transformation.

Chiral invariant terms that describe Goldstone boson selfinteractions are constructed from the fields in (2) and their derivatives.Hadrons containing a heavy quark emit and absorb light Goldstone bosons with noappreciable change in their four velocities. They are consequently described by velocitydependent fields.

In the meson sector, we introduce the operators Pi(v) and P ∗iµ(v) thatannihilate pseudoscalar and vector mesons with quark content Qq. When the suppressedheavy quark label carried by these fields corresponds to charm, their individual componentsare given by(P1, P2, P3) = (D0, D+, D+s )(P ∗1 , P ∗2 , P ∗3 ) = (D∗0, D∗+, D∗+s ).

(3)1 This introductory discussion closely follows that presented in refs. [4,5] to which we referinterested readers for further details.1

In the infinite quark mass limit, it is useful to combine the degenerate meson spin statesinto the 4 × 4 matrix field [1,6]Hi(v) = 1 + v/2−Pi(v)γ5 + P ∗iµ(v)γµ(4a)and its conjugateHi(v) = γ0H†γ0 =P †i(v)γ5 + P ∗†µi(v)γµ1 + v/2. (4b)H carries a heavy quark spinor index and a separate light antiquark spinor index andtransforms as an antitriplet under SU(3)L+R and doublet under SU(2)v.Baryons with quark content Qqq enter into the theory in two types depending upon theangular momentum of their light degrees of freedom.

In the first case, the light spectatorsare arranged in a symmetric spin-1 configuration that couples with the heavy spin- 12 quarkto form JP = 12+ and JP = 32+ states. When the heavy partner is taken to be charm,the spin- 12 states are destroyed by the Dirac operators appearing in the symmetric sextetrepresentationS =6XI=1SITI(6) =Σ++cq12Σ+cq12Ξ+c′q12Σ+cΣ0cq12Ξ0c′q12Ξ+c′q12Ξ0c′Ω0c.

(5)Their spin- 32 counterparts are annihilated by the corresponding Rarita-Schwinger field S∗µ.We again combine the Dirac and Rarita-Schwinger operators into the “super” fields [7]Sijµ (v) =r13(γµ + vµ)γ5Sij(v) + S∗µij(v)Sµij(v) = −r13Sij(v)γ5(γµ + vµ) + S∗ijµ(v). (6)Then Sµ transforms as a sextet under SU(3)L+R, doublet under SU(2)v, and is an axialvector.The spectators in the second case are bound together into an antisymmetric spin-0state.

The resulting JP = 12+ baryons are assigned to the field Ti(v), which is an SU(3)L+Rantitriplet and SU(2)v doublet. When Q = c, the components of Ti are the singly charmedbaryons(T1, T2, T3) = (Ξ0c, −Ξ+c , Λ+c ).

(7)2

These antitriplet baryons can alternatively be arranged into the antisymmetric matrixT =3Xi=1TiT(3)i=0q12Λ+cq12Ξ+c−q12Λ+c0q12Ξ0c−q12Ξ+c−q12Ξ0c0(8)whereT (3)ijk = ǫijk/√2.We can now construct the zeroth order effective chiral Lagrangian that describesthe low energy interactions between light Goldstone bosons and heavy hadrons in theinfinite heavy quark mass limit.The leading order terms must be hermitian, Lorentzinvariant, light flavor and heavy quark spin symmetric, and parity even.We can alsoreadily incorporate electromagnetism into the hybrid chiral theory by gauging a U(1)EMsubgroup of the global SU(3)L × SU(3)R symmetry group. Only long wavelength photonswith energies less than the chiral symmetry breaking scale explicitly remain in the lowenergy theory while short wavelength modes are integrated out.

In d = 4 −ǫ dimensions,the effective Lagrangian looks like 2L(0) = −14F µνFµν + µ−ǫf 24Tr(DµΣ†DµΣ)(9a)L(0)v=XQ=c,bn−iTrH′iv · DH′i−iSµijv · DSijµ + (MS −MT)SµijSijµ + iTiv · D Ti+ g1TrH′i(A/)ijγ5H′j+ ig2εµνσλSµikvν(Aσ)ij(Sλ)jk+ g3hǫijkTi(Aµ)jl Sklµ + ǫijkSµkl(Aµ)ljTiio. (9b)The Goldstone bosons explicitly couple to the matter fields through the axial vector com-binationAµ = i2(ξ†Dµξ −ξDµξ†).

(10a)They also communicate via the vector fieldVµ = 12(ξ†Dµξ + ξDµξ†)(10b)2 Meson contributions are written in terms of the dimension- 32 field H′ = √MHH so that allheavy mass dependence is removed from the leading order Lagrangian.3

that appears inside the heavy hadron covariant derivatives 3DµH′i = ∂µH′i −H′j(V µ)ji −iµǫ/2eAµQQH′i −H′jQjiDµSijν = ∂µSijν + (V µ)ikSkjν + (V µ)jkSikν −iµǫ/2eAµQQSijν + QikSkjν + QjkSikνDµTi = ∂µTi −Tj(V µ)ji −iµǫ/2eAµQQTi −TjQji. (11)The remaining Goldstone covariant derivatives in (9a) and (10) are given byDµΣij = ∂µΣij −iµǫ/2eAµQikΣkj −ΣikQkjDµξij = ∂µξij −iµǫ/2eAµQikξkj −ξikQkj(12)whereQ =Q1Q2Q3=23−13−13(13)denotes the light quark electric charge matrix.Spin symmetry violating contributions to the chiral Lagrangian enter at O(1/mQ).Among these are heavy quark magnetic moment terms which mediate M1 radiative tran-sitions.

As we will see, these terms are completely fixed by heavy quark number conser-vation. This simple but crucial observation allows one to use experimental meson decayinformation to determine the parameter g1.Recall that the photon gauge field couples to the conserved current that counts heavyquark number in the underlying QCD theory as well as in the low energy HQET andHHCPT.

The original QCD current appears in its well-known Gordon decomposed formasJQCDµ= Q(p′)γµQ(p) =12mQQ(p′)(p′ + p)µ + iσµν(p′ −p)νQ(p). (14)Running down in energy to the heavy quark thresholds and invoking the velocity super-selection rule [8] to set p(′) = mQv + k(′), one can match this tree level current onto thecorresponding HQET expressionJHQETµ= h(Q)vvµ +i2mQ(−→∂µ −←−∂µ) +12mQσµν(−→∂ν + ←−∂ν)h(Q)v.(15)3 We distinguish the photon field Aµ from the axial vector Goldstone current Aµ by writingthe former in calligraphy type.

Similarly, we let Q represent the electric charge operator, whichis different from the heavy quark symbol Q.4

The Aµ gauge field couples to this current in the LagrangianL(HQET)v=XQ=c,bnh(Q)v(iv·D)h(Q)v+ a1O1 + a2O2 + a3O3o(16)where the O(1/mQ) Oi operators are constructed from either symmetric or antisymmetriccombinations of two HQET covariant derivatives Dµ = ∂µ −iµǫ/2gGµaTa −iµǫ/2eAµQ:O1 =12mQh(Q)v(iD)2h(Q)v(17a)O2 = µǫ/2g4mQh(Q)vσµνTah(Q)vGµνa(17b)O3 = µǫ/2eQQ4mQh(Q)vσµνh(Q)vF µν. (17c)The ai coefficients of these dimension-five operators are thus fixed by (15) and equal unityat lowest order [9].Running down further in energy from the heavy quark thresholds to the chiral sym-metry breaking scale Λχ, we match the electromagnetic pieces of operators O1 and O3 ontothe following short distance contributions to the HHCPT Lagrangian:L(short)v=XQ=c,bn−12mQTr(H′(iD)2H′) −µǫ/2QQe(mQ)4mQTr(H′σµνH′F µν)−12mQSλij(iD)2Sijλ −µǫ/2QQe(mQ)4mQSλijσµνSijλ F µν+12mQTi(iD)2Ti + µǫ/2QQe(mQ)4mQTiσµνTiF µνo.

(18)These O(1/mQ) terms describe the interaction of photons with the heavy quark constituentinside a H′, S or T hadron. Consequently, the Lorentz and flavor indices for the hadrons’light degrees of freedom are trivially contracted.

Heavy quark number conservation deter-mines the ratio of the operator coefficients in (18) to the kinetic terms in the zeroth orderLagrangian (9b). The conserved HQET current therefore matches ontoJHHCP Tµ= −TrH′ihvµ +i2mQ(−→∂µ −←−∂µ) +12mQσµν(−→∂ν + ←−∂ν)iH′i−Sλijhvµ +i2mQ(−→∂µ −←−∂µ) +12mQσµν(−→∂ν + ←−∂ν)iSijλ+ Tihvµ +i2mQ(−→∂µ −←−∂µ) +12mQσµν(−→∂ν + ←−∂ν)iTi(19)5

in the low energy chiral theory.Photons also couple to the light brown muck inside heavy hadrons leaving the spinsof their heavy quark constituents unaltered.Such long distance interactions generateadditional electromagnetic contributions to the effective Lagrangian at the Λχ scale. Wefocus upon just the induced magnetic moment terms:L(long)v= µǫ/2e(Λχ)ΛχncHTrH′iH′j(−Q)jiσµνF µν+ icSSµ,ijQikSkjν + QjkSikνF µν+ cSThǫijkTivµQjl Sklν + ǫijkSν,klvµQljTiiF µνo.

(20)A few points about these long distance operators should be noted. Firstly, the suppressedheavy quark spinor indices in these spin symmetry preserving terms are simply contracted.Their light Lorentz and flavor indices on the other hand are nontrivially arranged.

Sec-ondly, the coefficients cH, cS and cST are a priori unknown. But naive dimensional analysissuggests that they are of order one [10].

Finally, there is no long distance magnetic mo-ment interaction for just the antitriplet baryon since the photon field cannot couple to itsspinless light degree of freedom.Having identified the short and long distance magnetic moment terms in the low energychiral theory, we can now study M1 radiative transitions between meson and baryon states.Since the hyperfine splitting between charmed pseudoscalar and vector meson partners isonly slightly greater than a pion mass, the electromagnetic decay D∗→Dγ competes withthe strong process D∗→Dπ. The greater phase space for the electromagnetic transitionoffsets its inherently smaller amplitude.Bottom vector mesons must radiatively decaybecause pion emission is kinematically forbidden.

So these M1 meson processes are ofgenuine phenomenological interest. Similar considerations apply to the baryon transitions.The vector meson and sextet baryon radiative decay rates are readily determined fromthe magnetic moment terms in (18) and (20):Γ(P ∗i →Piγ) =(21a)23 MPMP ∗M 2P ∗−M 2PMP ∗3h QQ4mQαEM(mQ)1/2 + cHΛχQiαEM(Λχ)1/2i2Γ(S∗I →SIγ) =(21b)118 MSMS∗M 2S∗−M 2SMS∗3h QQmQαEM(mQ)1/2 + 2 cSΛχTr(TI(6)†QTI(6))αEM(Λχ)1/2i2Γ(S(∗)I →Tjγ) =(21c)16 MTMS(∗)M 2S(∗) −M 2TMS(∗)3hcSTΛχTr(T(3)j†QTI(6))αEM(Λχ)1/2i2.6

One can clearly identify the short and long distance contributions to these partial widthsfrom their electric charges and associated inverse mass scales. The corresponding stronginteraction decay rates are derived from the Goldstone axial vector couplings in the leadingorder Lagrangian (9b):Γ(P ∗i →Pjπa) =(22a)g2148πf 2 MPMP ∗"[M 2P ∗−(MP + mπ)2][M 2P∗−(MP −mπ)2]M 2P∗#3/2|(T a)ji|2Γ(S∗I →SJπa) =(22b)g22144πf 2 MSMS∗"[M 2S∗−(MS + mπ)2][M 2S∗−(MS −mπ)2]M 2S∗#3/2|Tr(TJ(6)†T aTI(6))|2Γ(S(∗)I →Tjπa) =(22c)g2324πf 2 MTMS(∗)"[M 2S(∗) −(MT + mπ)2][M 2S(∗) −(MT −mπ)2]M 2S(∗)#3/2|Tr(T(3)j†T aTI(6))|2.None of the heavy hadron electromagnetic and strong partial widths have been directlymeasured.

However, values for D∗branching fraction ratios are known [11]:R0γ = Γ(D∗0 →D0γ)Γ(D∗0 →D0π0) = 0.572 ± 0.057 ± 0.081R+γ = Γ(D∗+ →D+γ)Γ(D∗+ →D+π0) = 0.035 ± 0.047 ± 0.052. (23)Taken in conjunction with the isospin relationR+π = Γ(D∗+ →D0π+)Γ(D∗+ →D+π0) = 2.21 ± 0.07,(24)these data yield the following branching fractions: 4D∗+ →D0π+68.1 ± 1.0 ± 1.3%(25a)D∗+ →D+π030.8 ± 0.4 ± 0.8%(25b)D∗+ →D+γ1.1 ± 1.4 ± 1.6%(25c)D∗0 →D0π063.6 ± 2.3 ± 3.3%(25d)D∗0 →D0γ36.4 ± 2.3 ± 3.3%.

(25e)4 These very recent CLEO values differ significantly from Particle Data Group world averages[12].7

Using the branching fraction ratios for the two independent D∗charge modes in (23),we can deduce the parameters cH/Λχ and g21 that enter into the heavy meson electromag-netic and strong decay rates respectively. To extract these unknown couplings from the D∗data and to predict the B∗widths, we must specify numerical values for the charm and bot-tom mass parameters mc and mb.

Since these quark masses are sources of large theoreticaluncertainty, we perform the fit twice. First we assume (mc, mb) = (1500 MeV, 4500 MeV),and then we take (mc, mb) = (1700 MeV, 5000 MeV).

Reasonable estimates for the heavyquark masses are covered by the range between these two sets of input values.From the charm vector meson ratios, we find two equations for the two unknowns:g−21"12mcαEM(mc)1/2 −cHΛχαEM(Λχ)1/2#2=9128πf 2hM 2D∗+ −(MD+ + mπ0)2M 2D∗+ −(MD+ −mπ0)2i3/2M 2D∗+ −M 2D+3R+γg−21"14mcαEM(mc)1/2 + cHΛχαEM(Λχ)1/2#2=9512πf 2hM 2D∗0 −(MD0 + mπ0)2M 2D∗0 −(MD0 −mπ0)2i3/2M 2D∗0 −M 2D03R0γ. (26)Following the suggestion of naive dimensional analysis, we choose the roots of thesequadratic equations that yield values for g21 of order unity.

The results of the parame-ter fit are then listed as functions of the charm quark mass in Table 1:Couplingmc = 1500 MeVmc = 1700 MeVcH/Λχ(−0.68 ± 0.50)/(1000 MeV) (−0.60 ± 0.44)/(1000 MeV)g210.43 ± 0.610.34 ± 0.48Table 1These results only weakly depend upon Λχ through the logarithmic running of the finestructure constant.Therefore, a very precise numerical value for the chiral symmetry8

breaking scale need not be specified. However, if one reasonably assumes Λχ ≈1000 MeV,then the value for cH turns out to be of order one and is consistent with our earlier ex-pectations.

We also note for comparison that the nonrelativistic quark model estimate forthe squared Goldstone axial vector parameter is 0.7 <∼g21 <∼1.0 [3,13]. The HHCPT centralvalue for this coupling is therefore of the same order of magnitude but smaller than thequark model number.

The large error bars on g21 reflect the 200% uncertainty in the mea-surement (25c) of the D∗+ →D+γ branching fraction. Improvements in the experimentalvalue will yield more precise estimates for this basic chiral Lagrangian parameter.Having found cH/Λχ and g21, we can now obtain model independent predictions for thetotal and partial widths of all D∗and B∗vector mesons.

Our predictions are summarizedin Table 2:9

Width ( MeV)mc = 1500 MeVmc = 1700 MeVmb = 4500 MeVmb = 5000 MeVΓ(D∗+)(12.44 ± 12.27) × 10−2(9.70 ± 9.56) × 10−2Γ(D∗+ →D+π0)(3.56 ± 5.06) × 10−2(2.77 ± 3.94) × 10−2Γ(D∗+ →D0π+)(7.83 ± 11.13) × 10−2(6.10 ± 8.68) × 10−2Γ(D∗+ →D+γ)(1.06 ± 1.05) × 10−2(0.83 ± 0.81) × 10−2Γ(D∗0)(6.49 ± 7.94) × 10−2(5.06 ± 6.19) × 10−2Γ(D∗0 →D0π0)(5.36 ± 7.63) × 10−2(4.18 ± 5.94) × 10−2Γ(D∗0 →D0γ)(1.13 ± 2.20) × 10−2(0.88 ± 1.71) × 10−2Γ(B∗+) = Γ(B∗+ →B+γ)(8.46 ± 11.94) × 10−4(6.60 ± 9.31) × 10−4Γ(B∗0) = Γ(B∗0 →B0γ)(1.63 ± 2.61) × 10−4(1.27 ± 2.03) × 10−4Table 2Current upper bounds on D∗widths are about an order of magnitude greater than thecentral values quoted here, while no B∗decay information is yet available. Comparison ofthese theoretical results with experimental data must therefore be left for the future.To conclude, we comment upon several possible extensions of this work.

In the mesonsector, a number of refinements of our leading order analysis should be pursued. Perturba-tive QCD corrections, subleading O(1/mQ) and SU(3)L+R breaking effects, and calculable10

nonanalytic terms from Goldstone boson loop diagrams may all be systematically incorpo-rated into the HHCPT framework to yield improved values for the meson parameters anddecay rates. D∗s and B∗s decays can also be worked out and studied in a straightforwardfashion.

For the sextet baryons, the present absence of branching ratio data precludesour determining the baryon couplings (cS/Λχ, g22) and (cST/Λχ, g23) as well as the widthsof the spin- 32 states in precisely the same manner as their meson analogues. Nonetheless,such baryon data will eventually become available in the future.

So the enhancementsmentioned above for the mesons ought to be carried out for the baryons as well. Finally,the scope of HHCPT can be broadened to include higher resonances such as the D1 andD∗2 states [14].

Electromagnetic interactions for these meson and baryon excitations maybe incorporated into the theory along the same lines as those for the heavy hadron H′, Sand T ground states.AcknowledgementsWe thank Glenn Boyd and Mark Wise for several discussions and for communicatingresults prior to publication. These authors and their collaborators have independentlyderived many of the findings reported here [15].

We are also grateful to Steve Schaffner,Mat Selen and Hitoshi Yamamoto for providing access to CLEO data. Finally, PC thanksthe theory group at Fermilab where part of the work on this letter was performed for theirwarm hospitality.

This work was supported in part by the National Science Foundationunder contract PHY-87-14654 and by the Texas National Research Commission underGrant # RGFY9206.11

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