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CALT-68-1912 DOE RESEARCH AND DEVELOPMENT REPORT Strong and Electromagnetic Decays ∗ of Two New Λ Baryons c 4 9 9 1 n Peter Cho† a J Lauritsen Laboratory 8 California Institute of Technology 1 Pasadena, CA 91125 1 v 6 7 2 Abstract 1 0 4 Two recently discovered excited charm baryons are studied within the framework 9 / of Heavy Hadron Chiral Perturbation Theory. We interpret these new baryons which h p lie 308 MeV and 340 MeV above the Λ as I = 0 members of a P-wave spin doublet. - c p e Differential and total decay rates for their double pion transitions down to the Λc ground h : state are calculated. Estimates for their radiative decay rates are also discussed. We v i find that the experimentally determined characteristics of the Λ∗ baryons may be simply X c r understood in the effective theory. a 1/94 † Work supported in part by an SSC Fellowship and by the U.S. Dept. of Energy under DOE Grant no. DE-FG03-92-ER40701. 1. Introduction The discovery of the first excited charm baryon has recently been announced by the ARGUS, CLEO and E687 groups [1–3]. The new state lies approximately 340 MeV above the Λ (2286 MeV) and decays to it via double pion emission. Although its spin, isospin c and parity are not yet known, this new charmed baryon has been preliminarily interpreted as a Λ∗ resonance. CLEO has further reported evidence for a second Λ∗ excitation at c c 308 MeV above Λ [4]. The second resonance also decays through a double pion mode c that is consistent with the two step process Λ∗ Σ π followed by Σ Λ π. In contrast, c → c c → c CLEO finds no evidence for an intermediate Σ in the decay of the first Λ∗ excitation [2]. c c In this article, we will analyze these new baryon states and their dominant decay modes within the framework of Heavy Hadron Chiral Perturbation Theory (HHCPT). This hybrid effective theory represents a synthesis of Chiral Perturbation Theory and the Heavy Quark Effective Theory (HQET) and describes the low energy interactions between light Goldstone bosons and hadrons containing a heavy quark [5–9]. Since its development afewyearsago, HHCPThasprimarilybeenappliedtothestudyofgroundstatecharmand bottom hadrons. Ground state mesons and baryons are more tightly restricted by heavy quark spin symmetry than their excited counterparts. Moreover, experimental information has been much more sparse for the latter than the former. It is therefore not surprising that theorists have concentrated upon the lowest lying hadrons in the past. Now however that new data is being collected, it is worthwhile to broaden the scope of HHCPT and incorporate excited heavy hadrons into the effective theory. The first excited heavy mesons and baryons are P-wave hadrons that carry one unit of orbital angular momentum. P-wave mesons have already been investigated within the HHCPT framework [10–12]. It is straightforward to extend the formalism and include P- wave baryons as well. A number of unknown couplings enter into the excited baryon sector which limits one’s predictive power. But as we shall see, all the general characteristics of the two Λ∗ baryons reported by ARGUS, CLEO and E687 are consistent with their c being members of an excited spin symmetry doublet. Although our findings will be more qualitative than quantitative, we hope this work may help guide experimentalists as they continue to study these new charmed baryons. Our paper is organized as follows. In section 2, we incorporate the lowest lying excited baryon doublet into the heavy baryon chiral Lagrangian. We then focus upon the two new Λ∗ members of this doublet and analyze their strong interaction decays in section 3. c Radiative transitions are discussed in section 4. Finally, we close in section 5 with some thoughts on future directions for investigation. 1 2. The Heavy Baryon Chiral Lagrangian We begin by recalling some basic aspects of the baryon sector in Heavy Hadron Chiral Perturbation Theory [7,8]. Ground state baryons with quark content Qqq have zero orbital angular momentum and occur in two types depending upon the angular momentum j of ℓ their light degrees of freedom. In the first case, the light brown muck is arranged in a symmetric j = 1 configuration which transforms as a sextet under flavor SU(3). The ℓ spectators consequently couple with the heavy quark to form JP = 1+ and JP = 3+ 2 2 S-wave bound states. When the heavy quark is taken to be charm, the spin-1 states are 2 annihilated by velocity dependent Dirac operators Sij(v) whose individual components are given by 1 S11 = Σ++ S12 = Σ+ S22 = Σ0 c r2 c c 1 1 (2.1) S13 = Ξ+′ S23 = Ξ0′ r2 c r2 c S33 = Ω0. c Their spin-3 counterparts are destroyed by corresponding S∗ij(v) Rarita-Schwinger op- 2 µ erators. In the second case, the light degrees of freedom form an antisymmetric j = 0 ℓ combination which transforms as a flavor antitriplet. Coupling with the heavy quark then yields JP = 1+ baryons which we associate with the field T (v). When Q = c, the indi- 2 i vidual components of T are the singly charmed baryons i T = Ξ0 T = Ξ+ T = Λ+. (2.2) 1 c 2 − c 3 c The complete spectrum of the first orbitally excited P-wave Qqq baryons is quite complicated. The lowest lying such hadrons correspond to bound states that have one unit of orbital angular momentum inserted between the heavy quark and light diquark pair. In this case, spin statistics constrain the light degrees of freedom to belong to either a j = 1 multiplet which transforms as a flavor antitripletor else to j = 0, 1 or 2 multiplets ℓ ℓ which transform as flavor sextets. Nonrelativistic quark model calculations indicate that the antitriplet multiplet is isolated and lies significantly below all other P-wave states [13]. We will therefore only incorporate this lightest j = 1 multiplet into the chiral Lagrangian. ℓ We assign the Dirac and Rarita-Schwinger operators R (v) and R∗ (v) to its JP = 1− and i µi 2 JP = 3− states. As we shall see, the two newly discovered Λ∗ baryons are well described 2 c as the I = 0 members of R and R∗. µ 2 In the infinite heavy quark mass limit, it is useful to combine together the degenerate J = 1 and J = 3 members of the ground state sextet and excited antitriplet multiplets 2 2 into the baryon “superfields” [8,14] 1 = (γ +v )γ5R +R∗ Rµi r3 µ µ i µi (2.3) 1 ij = (γ +v )γ5Sij +S∗ij. Sµ r3 µ µ µ The superfield for the ground state antitriplet baryons is simply identical to T . The i i T superfields transform under parity as (~x,t) P γ µ( ~x,t) µ 0 R −→ R − (~x,t) P γ µ( ~x,t) (2.4) µ 0 S −→ − S − (~x,t) P γ ( ~x,t) 0 T −→ T − and obey the constraints 1+v/ 1+v/ = = 1+v/ µ µ µ µ 2 R R 2 S S = . (2.5) 2 T T vµ = 0 vµ = 0 µ µ R S These conditions ensure that and contain six degrees of freedom while has two. µ µ R S T The degree of freedom count thus agrees with the number of states that the superfields represent [16]. The constraints in (2.5) also fix the shifts in the baryon superfields induced by the reparameterization transformation v v +ǫ/M → (2.6) k k ǫ → − where v ǫ = 0. This change of variables leaves invariant the total four-momentum p = · Mv+k of a heavy hadron and induces only an O(1/M2) correction to v2 = 1. The method for determining the induced shifts in the baryon superfields is entirely analogous to that for their meson counterparts which has previously been discussed in ref. [12]. So we only quote the results here: /ǫ ǫν ν δ = R v µ µ µ R 2MR − M /ǫ ǫν δ = Sνv (2.7) µ µ µ S 2MS − M /ǫ δ = . T 2MT 3 The requirement that the effective theory remain invariant under the transformations in (2.6) and (2.7) forbids certain terms from appearing in the chiral Lagrangian [15]. The heavy baryons in the , and multiplets can interact with one another via µ µ R S T emission and absorption of light Goldstone bosons. The Goldstone bosons result from the spontaneous breaking of SU(3) SU(3) chiral symmetry down to its diagonal SU(3) L× R L+R flavor subgroup and appear in the pion octet 1π0 + 1η π+ K+ 2 6 8 1 q q  πππ = πaTa = π− 1π0 + 1η K0 . (2.8) Xa=1 √2  −q2 0 q6   K− K 2η  −q3  It is convenient to arrange these fields into the exponentiated matrix functions Σ = e2iπππ/f and ξ = eiπππ/f where the parameter f equals the pion decay constant f = 93 MeV at π lowest order. The matrix functions transform under the chiral symmetry group as Σ LΣR† → (2.9) ξ LξU†(x) = U(x)ξR† → where L and R represent global elements of SU(3) and SU(3) while U(x) acts like a L R local SU(3) transformation. We further define the vector and axial vector fields L+R 1 1 1 Vµ = (ξ†∂µξ +ξ∂µξ†) = [πππ,∂µπππ] πππ, πππ,[πππ,∂µπππ] +O(πππ6) 2 2f2 − 24f4 h i (cid:2) (cid:3) (2.10) i 1 1 Aµ = (ξ†∂µξ ξ∂µξ†) = ∂µπππ + πππ,[πππ,∂µπππ] +O(πππ5) 2 − −f 6f3 (cid:2) (cid:3) which transform inhomogeneously and homogeneously under SU(3) respectively: L+R Vµ UVµU† +U∂µU† → (2.11) Aµ UAµU†. → The pions in (2.8) derivatively couple to the baryon matter fields via these vector and axial vector combinations. It isstraightforward to construct the lowest order effective Lagrangianwhich describes the low energy interactions between the Qqq baryons and Goldstone bosons. One simply 4 writes down all possible terms that are Lorentz invariant, light chiral and heavy quark spin symmetric, and parity even: (0) = i iv +∆M µ + µ iv +∆M Sij + iiv Lv Rµ − ·D R Ri Sij − ·D S µ T ·DTi QX=c,bn (cid:0) (cid:1) (cid:0) (cid:1) +ig ε µ vν(Aσ)i( λ)jk +ig ε µivν(Aσ)j( λ) 1 µνσλSik j S 2 µνσλR i R j (2.12) +h ǫ i(Aµ)j kl +ǫijk µ (A )l 1 ijkT lSµ Skl µ jTi h i +h ǫ µiv Aj kl +ǫijk µ v Al . 2 ijkR · lSµ Skl · jRµi h io A few points about this zeroth order Lagrangian should be noted. Firstly, the common mass splitting between the excited and ground state antitriplet multiplets is absorbed into the parameter ∆M = M M . Similarly, ∆M = M M represents the splitting R R T S S T − − between thegroundstatesextetandantitripletmultiplets. These parametersdo notvanish inthe zero orinfinite heavy quark mass limitsand therefore appropriately reside within the leadingorderchiral Lagrangian. Secondly, thecoupling constantsg andh in(2.12)are 1,2 1,2 expected to be of order unity on general dimensional analysis grounds [17]. However, their precise numerical values are a priori unknown and must be fitted to experiment. Finally, we observe that there are no terms in (2.12) which mediate the single Goldstone boson transitions R(∗) Tπππ and T Tπππ. Such processes violate heavy quark spin symmetry → → and occur only at next-to-leading order in the 1/m expansion. Q The current experimental status of the baryons appearing in the heavy hadron chiral Lagrangian is very uneven. Data on strange charmed baryons is in short supply, and several have not yet been discovered. In contrast, a number of experiments within the past year have filled in most of the nonstrange members of the antitriplet and sextet multiplets. We will therefore focus upon the zero strangeness baryons in the remainder of this work. (∗) (∗) TheenergylevelsoftheobservedΛ andΣ statesin , and areillustratedin c c µ µ R S T fig. 1. As indicated in the figure, we interpret the two recently observed excited charmed baryons as the I = 0 members of the multiplet. In the absence of well-established µ R names for these baryons, we adopt the nomenclature convention of ref. [18] and denote the JP = 1− and JP = 3− states as Λ and Λ∗ respectively. Averaging over the AR- 2 2 c1 c1 GUS, CLEO and E687 values for their masses, we find that they lie 308.0 2.0 MeV and ± 341.4 0.4 MeV above Λ . The splitting between these two P-wave baryon masses is com- c ± parable in magnitude to that between their P-wave meson analogues D (2421 MeV) and 1 D (2465 MeV). We will keep track of this phenomenologically important mass difference 2 even though it represents an O(1/m ) effect. c 5 The splitting between Σ∗ and Λ displayed in fig. 1 comes from another recent exper- c c imental result. The SKAT group claims to have observed the JP = 3+ Σ∗++ baryon for 2 c the first time in their bubble chamber experiment which uses a broad-band neutrino beam [19]. While their mass finding MΣ∗ = 2530 7 MeV must be treated with caution until c ± independently confirmed by another group, we will adopt their reasonable central value in our subsequent analysis. Fortunately, none of our results will sensitively depend upon the precise numerical value for the Σ∗ mass. c Having set up the necessary machinery for studying the two new Λ∗ baryons, we c proceed to examine their strong and radiative decay modes in the following two sections. 3. Strong Decays of ΛΛΛ∗∗∗ ccc The strong decays of the newly discovered excited charmed baryons are well-suited for Chiral Perturbation Theory analysis. The relatively small masses of Λ and Λ∗ above c1 c1 Λ kinematically restrict their strong decays to soft pion emission. We therefore expect c the chiral Lagrangian derivative expansion to be well-behaved for these new particles. (∗) Moreover, isospin conservation forbids single pion transitions between Λ and Λ . The c1 c excited I = 0 baryons must instead decay via an intermediate I = 1 state down to the I = 0 ground state. The released energy M M is thus shared by two pions. 1 Λ(∗) − Λc c1 Angular momentum and parity considerations require single pion transitions between the and multipletstogothroughL = 0orL = 2partialwaves. TheD-wavecoupling µ µ R S arises from dimension-five operators in the next-to-leading order chiral Lagrangian whose effects are quite suppressed. The S-wave coupling on the other hand is implemented by the dimension-four term proportional to h in (2.12) which links Λ with Σ and Λ∗ with 2 c1 c c1 Σ∗. The h operator consequently mediates the barely allowed transition Λ Σ π at c 2 c1 → c the rate h2 M Γ(Λ Σ π) = 2 Σc (M M )2 (M M )2 m2. (3.1) c1 → c 4πf2 M Λc1 − Σc Λc1 − Σc − π Λc1 p This process occurs so close to threshold that small isospin violating mass differences between members of the pion and charmed Sigma baryon multiplets cannot be ignored in 1 TheanalogouskinematicsforexcitedP-wavemesonsismuchlessfavorable. Forexample,the splitting between the D and D mesons is almost 600 MeV, and single pion transitions between 2 these two states are allowed. The validity of lowest order Chiral Perturbation Theory in this case is dubious at best. 6 the phase space factors of (3.1). Using the values M = 2452.0 MeV, M = 2453.4 MeV Σ0c Σ+c and M = 2453.1 MeV [20], we find the partial widths Σ++ c Γ(Λ+ Σ0π+) = 3.3h2 MeV (3.2a) c1 → c 2 Γ(Λ+ Σ+π0) = 6.0h2 MeV (3.2b) c1 → c 2 Γ(Λ+ Σ++π−) = 1.4h2 MeV. (3.2c) c1 → c 2 The analogous single pion transitions between the J = 3 baryons in and are 2 Rµ Sµ kinematically forbidden. Double pion decays of Λ and Λ∗ down to the Λ ground state proceed at leading c1 c1 c order via the two pole graphs displayed in fig. 2. In order to obtain convergent decay rates from these diagrams, we must take into account the nonzero widths h2 M Γ = 1 Λc (M M )2 m2 3/2 2.5h2 MeV Σc 12πf2 M Σc − Λc − π ≃ 1 Σc(cid:2) (cid:3) (3.3) h2 M ΓΣ∗c = 12π1f2 MΣΛ∗cc(cid:2)(MΣ∗c −MΛc)2 −m2π(cid:3)3/2 ≃ 24h21 MeV of the intermediate Σ and Σ∗ resonances. Their propagators thus appear as c c i D = Λ Σc v k (M M )+iΓ /2 + · − Σc − Λc Σc (3.4) i µν µν D = Λ Σ∗c v ·k −(MΣ∗c −MΛc)+iΓΣ∗c/2 + where Λ = (1+v/)/2 and Λµν = gµν +vµvν + 1(γµ +vµ)(γν vν) Λ denote spin-1 + + − 3 − + 2 (cid:2) (cid:3) and spin-3 projection operators respectively. We must also include a symmetry factor 2 of 1/2 in the angular integration over the pions’ momenta to avoid double counting the two identical bosons in the final state. A straightforward computation then yields the dimensionless differential decay rate dΓ Λ(∗) Λ πaπb δab h h 2 M c1 → c = 1 2 Λc (E2 m2)(E2 m2) (cid:0) dE1 (cid:1) 192π3(cid:16) f2 (cid:17) MΛ(∗)q 1 − π 2 − π c1 (3.5) E2(E2 m2) (E2 m2)E2 1 2 − π + 1 − π 2 ×(cid:20) M M E 2 +Γ2 /4 M M E 2 +Γ2 /4(cid:21) Λc(∗1) − Σc(∗) − 1 Σc(∗) Λc(∗1) − Σc(∗) − 2 Σc(∗) (cid:0) (cid:1) (cid:0) (cid:1) 7 expressed in terms of the two pion energies E and E = M M E measured in 1 2 Λ(∗) − Λc − 1 c1 the decaying body’s rest frame. 2 Integrating over E , we obtain the total rate 1 Γ Λ(∗) Λ πaπb = h22δab MΣc(∗) I (3.6) c1 → c 8π2f2 M 3/2 (cid:0) (cid:1) Λc(∗1) (cid:20)(MΣc(∗) −MΛc)2 −m2π(cid:21) where I = ΓΣc(∗) MΛc(∗1)−MΛc−mπ dE (E2 m2)(E2 m2) 2 Z 1q 1 − π 2 − π mπ E2(E2 m2) (E2 m2)E2 1 2 − π + 1 − π 2 . ×(cid:20) 2 2 (cid:21) M M E +Γ2 /4 M M E +Γ2 /4 (cid:16) Λc(∗1) − Σc(∗) − 1(cid:17) Σc(∗) (cid:16) Λc(∗1) − Σc(∗) − 2(cid:17) Σc(∗) (3.7) Since we do not know the values of h and h , we cannot extract precise quantitative 1 2 predictions from eqns. (3.5) (3.7). However, these formulas do provide useful qualitative insight into the P-wave baryons’ strong decays. In fig. 3, we plot h−2dΓ(Λ(∗) 2 c1 → Λ π0π0)/dE versus E with h set equal to unity. As can clearly be seen in the figure, c 1 1 1 Λ Λ π0π0 is dominated by the pole regions where the intermediate Σ+ state is very c1 → c c close to being on-shell. Its integrated rate is thus well approximated by the single π0 partial width in (3.2b). The rate in the charged pion channel is similarly well approximated by the sum of the widths in (3.2a) and (3.2c). Indeed, evaluating the phase space integral in (3.7) using the narrow width approximation Γ /2 Σc πδ(M M E), (3.8) (M M E)2 + Γ /2 2 ≃ Λc1 − Σc − Λc1 − Σc − Σc (cid:0) (cid:1) we simply recover eqn. (3.1) for Γ(Λ Σ π) which is independent of coupling constant c1 c → h . 1 Nonresonant contributions generate a slight dependence of Γ(Λ Λ πaπb) upon h c1 c 1 → as shown in fig. 4. But the decay of the JP = 1− state may essentially be viewed as the 2 two step process Λ Σ π followed by Σ Λ π. In contrast, the double pion decay c1 c c c → → of Λ∗ cannot be regarded as a sequential transition. The virtual Σ∗ intermediate state c1 c 2 In the infinite charm mass limit,the recoilingΛc baryon carries off momentum but no kinetic energy. The two pions thus share all of the energy released by the decaying Λ(∗). This situation c1 is similar to bouncing a ball off the earth. The earth must recoil to conserve momentum, but the ball bounces back with practically all its original kinetic energy [21]. 8 is very much off-shell and produces no large resonant contribution to Λ∗ Λ ππ. As a c1 → c result, the strong interaction partial width of the JP = 3− state is more than an order of 2 magnitude smaller than that of its JP = 1− partner. 2 As advertised in the Introduction, these qualitative findings on the excited charm baryon decay modes are in basic accord with the recent CLEO results reported in refs. [2] and [4]. They thus bolster one’s confidence in the interpretation of the two new states as Λ∗ baryons. To make further progress however, we need width information to pin down c the values of the coupling constants in chiral Lagrangian (2.12). ARGUS has set a 90% CL upper bound of 3.2 MeV on the width of Λ∗ [1]. Unfortunately, this limit places c1 only a weak constraint on the allowed parameter space in the h h plane. As fig. 4 1 2 demonstrates, the true natural width of Λ∗ is most likely too narrow to be resolved by c1 current experimental detectors. On the other hand, there is a much better chance that the Λ resonance is wide enough to be measured. In the I = 1 sector, the width resolving c1 prospects for the JP = 1+ and JP = 3+ members of the doublet are just opposite 2 2 Sµ those for the JP = 1− and JP = 3− members of . We are therefore hopeful that 2 2 Rµ experimentalists will be able to fix some of the free parameters in the heavy baryon chiral Lagrangian in the near future. 4. Electromagnetic Decays of ΛΛΛ∗∗∗ ccc (∗) TheonlydecaymodesofthetwonewΛ baryonsthathavesofarbeenexperimentally c1 observed are their double pion transitions to Λ . But as shown in fig. 1, these P-wave c hadrons can also de-excite down to the ground state via single photon emission. Unlike the stronginteractionprocesses, theradiativechannelsarenotseverelyphasespacesuppressed. Moreover, they produce two rather than three bodies in the final state. So the inherently weaker strength of the electromagnetic transitions could be offset by their more favorable kinematics. We explore such a possibility in this section. Electromagnetic interactions may be incorporated into Heavy Hadron Chiral Per- turbation Theory by gauging a U(1) subgroup of the global SU(3) SU(3) chiral EM L × R symmetry group. All derivatives appearing in the velocity dependent effective Lagrangian are then promoted to covariant derivatives with respect to electromagnetism. The leading dimension-four operators in (2.12) cannot contribute to S-matrix elements between states containing real photons. So to study heavy meson and baryon radiative transitions, one 9