PHYSICAL REVIEW D 100, 114001 (2019)

Low-energy axial-vector transitions from decuplet to octet

Måns Holmberg and Stefan Leupold Institutionen för fysik och astronomi, Uppsala Universitet, Box 516, S-75120 Uppsala, Sweden

(Received 1 October 2019; published 2 December 2019)

Axial-vector transitions of decuplet to octet baryons are parametrized at low energies guided by a complete and minimal chiral Lagrangian up to next-to-leading order. It is pointed out that beyond the well- known leading-order term, there is only one contribution at next-to-leading order. This contribution is flavor symmetric. Therefore the corresponding low-energy constant can be determined in any sector. As functions of this low-energy constant, we calculate the decay widths and Dalitz distributions for the decays of decuplet baryons to octet baryons, , and and for the weak decay of the Omega to a cascade baryon, an , and an antineutrino.

DOI: 10.1103/PhysRevD.100.114001

I. INTRODUCTION AND SUMMARY In this way, one assumes a too large number of to be fitted parameters, and one misses cross relations between form One of the interesting processes of - factors and also between different processes. Therefore we scattering is the production of the lowest-lying Delta state have decided to go one step back and analyze the NLO [1–5]. On the theory side, the relevant quantities are the structure, i.e., tree-level structure of the axial-vector tran- vector and axial-vector form factors for the transition of a sition form factors. Of course, this can only be the first step nucleon to a Delta. While the vector transition form factors of a more detailed investigation that should go beyond the can also be addressed in the electroproduction of a Delta on NLO level. a nucleon, information about the axial-vector transition Let us start with the introduction of pertinent axial-vector form factors is scarce. Of course, the situation is even worse transition form factors where we can already point out in the strangeness sector. At least, the decay width of the some short-comings of previous works. With B denoting a reaction Ω− → Ξ0e−ν¯ has been measured [6,7]. e -1=2 baryon from the nucleon octet and B a spin-3=2 The purpose of the present work is to study the low- baryon from the Delta decuplet, the axial-vector transitions energy limit of these axial-vector transition form factors can be written as and to point out various ways how to measure these quantities. Recently we have established a complete and 0 μ ¯ 0 Γμν minimal relativistic chiral Lagrangian of next-to-leading hB ðp ÞjjAjBðpÞi ¼ uνðp Þ uðpÞð1Þ order (NLO) for the three-flavor sector including the lowest 0 − lying baryon octet and decuplet states [8]. Based on this with q ¼ p p and Lagrangian, we will calculate various observables that μν μ ν 2 μν 2 depend on the low-energy constants (LECs) that enter Γ ¼ q q H0ðq Þþg H1ðq Þ the axial-vector transition form factors. μ ν μν 2 μα ν 2 þðγ q − =qg ÞH2ðq Þþiσ qαq H3ðq Þ: ð2Þ Dealing with a chiral baryon Lagrangian at NLO means that we carry out tree-level calculations. But why do we The advantage of the decomposition (1), (2) lies in the fact bother about such NLO tree-level results, if the state of the that contributions to the axial-vector transition form factors art seems to be one-loop calculations [2,3]? Though axial- H start only at the ith chiral order for i ¼ 1, 2, 3. This is vector transition form factors have been calculated at the i easy to see because the momentum q carried by the axial- one-loop level of chiral perturbation theory (χPT), the vector current is a small quantity of chiral order 1. In numerically fairly unknown tree-level contributions have particular, this implies that H3 does not receive contribu- often been formulated based on nonminimal Lagrangians. tions from leading order (LO) and from NLO. The “regular” contributions to H0 start also at third chiral order, but H0 receives a contribution at LO from a Goldstone- Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. pole term. Further distribution of this work must maintain attribution to For the following rewriting, it is of advantage to recall the author(s) and the published article’s title, journal citation, the equations of motion for the spin-1=2 spinors u and the 3 and DOI. Funded by SCOAP . spin-3=2 Rarita-Schwinger vector-spinors [9] uν,

2470-0010=2019=100(11)=114001(7) 114001-1 Published by the American Physical Society MÅNS HOLMBERG and STEFAN LEUPOLD PHYS. REV. D 100, 114001 (2019)

0 0 ðp= − m8ÞuðpÞ¼0; ðp= − m10Þuνðp Þ¼0; ð3Þ TABLE I. Determination of hA from different decay channels of decuplet baryons. We used mixed states in the case of the Δ → Nπ decay. The errors come from the experimental uncertainties of where m8=10 denotes the mass of the considered baryon the masses, decay widths, and branching ratios. octet/decuplet state. Note that the mass difference m10 − m8 is counted as a small quantity. Decay hA The use of the form factor H3 is not very common. Δ → π 2 87 0 05 α 0 0 N . . Instead of iσμαq qν one often uses a structure pμqν − p · Σþ → Λπþ 2.39 0.03 qgμν; see e.g., [2,3] and references therein. The problem Σþ → ðΣπÞþ 2.2 0.1 with the latter structure is, however, that it looks like an Ξ0 → ðΞπÞ0 2.00 0.06 additional fourth independent term that contributes at NLO. Instead, the decomposition (2) shows that there are only — three structures up to and including NLO and H0 receives spatial ν. This selects the combination qja0 − q0aj, which only a Goldstone-boson pole contribution; i.e., there are constitutes an electric axial-vector field strength. independent only two terms up to and including NLO. The We close the present section by determining h . The 0 − 0 not A structure pμqν p · qgμν is required up to and includ- corresponding structure of the LO Lagrangian gives rise to ing NLO. To make contact between different ways how to decays of decuplet baryons to pions and octet baryons. parametrize the transition form factors, one can use a From each of the measured decay widths, one can extract Gordon-type identity, an estimate for hA. This is provided in Table I, which is in 0 α agreement with [11]. Note that this is an approximation u¯ νð−2pμ þðm10 þ m8Þγμ þ qμ − iσμαq Þu ¼ 0; ð4Þ for the decay widths that is accurate up to and including which can easily be established using the equations of NLO—because there are no additional contributions at motion (3). NLO [8]. One cannot expect to obtain always the very same We can now focus on the three form factors H0, H1 and numerical result for hA. But the spread in the obtained H2, which receive contributions already at LO or NLO, values can be regarded as an estimate for the neglected respectively. Using the Lagrangians of LO and NLO from contributions that appear beyond NLO. Indeed, those [8] one obtains contributions break the flavor symmetry. In Sec. II we present the basics of the relevant LO and 1 2 hA NLO Lagrangian of baryon octet plus decuplet and H0 q − ffiffiffi k ; ð Þ¼ p 2 − 2 f χ 2 q mGB Goldstone-boson octet PT. We also specify the numerical values of the previously determined LECs. This is the first hA H1 ¼ pffiffiffi k ; 2 f part of the paper. In Sec. III, we specify the relevant interaction terms of the Ω− → Ξ0e−ν¯ decay, and we also −2 e H2 ¼ cEkf ð5Þ present the LO and NLO decay width predictions (as a function of c ). Section IV is outlined similarly to Sec. III with the LO low-energy constant h and the NLO low- E A but in the case of BðJ ¼ 3=2Þ → BðJ ¼ 1=2Þγπ processes. energy constant c . The flavor factor k depends on E f We close by showing the c dependence of the single the considered channel, i.e., on the flavors of B, B and E differential decay width of Ξ0 → Ξ−πþγ and finally give the axial-vector current. Correspondingly, m denotes the GB some concluding remarks. mass of the Goldstone-boson that can be excited in the considered channel. The flavor factor is extracted from ϵade ¯ ν μ b c II. CHIRAL LAGRANGIAN Tabcða ÞdBe. We deduce from (5) the following information: In the The relevant part of the LO chiral Lagrangian for NLO approximation, one needs for all the axial-vector baryons including the spin-3=2 decuplet states is given decuplet-to-octet transitions only two flavor symmetric by [8,12–16] low-energy constants. It does not in which flavor channels one determines these constants. The main part of ð1Þ ¯ L ¼ trðBði=D − m 8 ÞBÞ this paper is devoted to suggestions how to pin down the baryon ð Þ ¯ μ γ α ν abc − γ ν abc NLO low-energy constant cE. þ Tabcði μναðD T Þ μνmð10ÞðT Þ Þ Obviously, cE contributes differently than hA. Thus a D ¯ γμγ F ¯ γμγ minimal and complete NLO Lagrangian must contain a þ 2 trðB 5fuμ;BgÞ þ 2 trðB 5½uμ;BÞ cE-type structure. It is missing, for instance, in [10].It h μ Affiffiffi ϵade ¯ b c ϵ ¯ e μ d abc might be interesting to explain why this low-energy con- þ p ð TabcðuμÞdBe þ adeBcðu ÞbTμ Þ stant has an index “E” for “electric”. At low energies, i.e., 2 2 H μ in the nonrelativistic limit for the baryons, the dominant − A ¯ γ γ ν c abd 0 Tabc ν 5ðu ÞdTμ ð6Þ contribution for the H2 form factor stems from γ and from 2

114001-2 LOW-ENERGY AXIAL-VECTOR TRANSITIONS FROM DECUPLET … PHYS. REV. D 100, 114001 (2019) ffiffiffi ffiffiffi 0 0 1 p p 1 with tr denoting a flavor trace. For the well-known π pffiffi η 2πþ 2 þ þ 3 K LO chiral Lagrangian is given by [17–19] B ffiffiffi ffiffiffi C p − 0 1 p 0 Φ B 2π −π pffiffi η 2 C ¼ @ þ 3 K A; 2 pffiffiffi pffiffiffi 1 Fπ − ¯ 0 2 Lð Þ μ χ 2K 2K − pffiffi η ¼ 4 trðuμu þ þÞ; ð7Þ 3 2 † † † u ≔ U ≔ expðiΦ=FπÞ;uμ ≔ iu ð∇μUÞu ¼ uμ: † † † with χ u χu uχ u and χ 2B0 s ip obtained ¼ ¼ ð þ Þ ð12Þ from the scalar and pseudoscalar sources s and p. The low- energy constant B0 is essentially the ratio of the light- The fields have the following transformation properties condensate and the square of the -decay constant; see, with respect to chiral transformations [12,19]: e.g., [18–21]. We have introduced the totally antisymmetrized products U → LUR†;u→ Luh† ¼ huR†; of two and three gamma matrices [22], † † uμ → huμh ;B→ hBh ; μ μ μν 1 μ ν μν abc a b c def ¯ † d † e † f ¯ γ ≔ γ γ − σ Tμ → h h h Tμ ; T → ðh Þ ðh Þ ðh ÞcT : 2 ½ ; ¼ i ð8Þ d e f abc a b def ð13Þ and In particular, the choice of upper and lower flavor indices is 1 used to indicate that upper indices transform with h under γμνα ≔ γμγνγα γνγαγμ γαγμγν 6 ð þ þ flavor transformations while the lower components trans- † μ α ν α ν μ ν μ α form with h . − γ γ γ − γ γ γ − γ γ γ Þ The chirally covariant derivative for a (baryon) octet is 1 γμν γα ϵμναβγ γ defined by ¼ 2 f ; g¼þi β 5; ð9Þ DμB ≔ ∂μB þ½Γμ;B; ð14Þ 0 1 2 3 respectively. Our conventions are γ5 ≔ iγ γ γ γ and ϵ0123 ¼ −1 (the latter in agreement with [22] but opposite for a decuplet T by to [15,23]). If a formal manipulation program is used to calculate spinor traces and Lorentz contractions a good μ abc ≔ ∂μ abc Γμ a a0bc Γμ b ab0c ðD TÞ T þð Þa0 T þð Þb0 T check for the convention for the Levi-Civita symbol is the μ c abc0 Γ 0 T ; 15 last relation in (9). þð Þc ð Þ The spin-1=2 octet baryons are collected in (Ba is the b for an antidecuplet by entry in the ath row, bth column) μ ¯ μ ¯ μ a0 ¯ μ b0 ¯ 0 1 0 1 1 ðD TÞ ≔ ∂ Tabc − ðΓ Þ Ta0bc − ðΓ Þ Tab0c pffiffi Σ pffiffi ΛΣþ abc a b 2 þ 6 p μ c0 ¯ B C − Γ 0 − 1 0 1 ð Þc Tabc ; ð16Þ B Σ − pffiffi Σ pffiffi Λ C B ¼ @ 2 þ 6 n A: ð10Þ − 0 2 Ξ Ξ − pffiffi Λ and for the fields by 6

∇μU ≔ ∂μU − iðvμ þ aμÞU þ iUðvμ − aμÞð17Þ The decuplet is expressed by a totally symmetric flavor abc tensor T with with 1 1 111 þþ 112 þ † T ¼ Δ ;T¼ pffiffiffi Δ ; Γμ ≔ ðu ð∂μ − iðvμ þ aμÞÞu 3 2 † 1 þ uð∂μ − iðvμ − aμÞÞu Þ; ð18Þ T122 ¼ pffiffiffi Δ0;T222 ¼ Δ−; 3 1 1 1 where v and a denote external vector and axial-vector T113 ¼ pffiffiffi Σþ;T123 ¼ pffiffiffi Σ0;T223 ¼ pffiffiffi Σ−; sources. 3 6 3 In (7), χ includes the mass term when replacing the 1 1 þ 133 0 233 − 333 scalar source s by the quark mass matrix. As for the baryon T ¼ pffiffiffi Ξ ;T¼ pffiffiffi Ξ ;T¼ Ω: ð11Þ 3 3 masses, mð8Þ [mð10Þ]in(6) denotes the masses of the octet (decuplet) baryons in the chiral limit. At NLO, octet and χ The Goldstone are encoded in decuplet flavor breaking terms appear (containing þ) that

114001-3 MÅNS HOLMBERG and STEFAN LEUPOLD PHYS. REV. D 100, 114001 (2019) are responsible for splitting the baryon masses such that The values of the low-energy constants bM;D=F and dM they are sufficiently close to the physical masses [8,24,25]. are determined by fitting the calculated and measured Therefore, we use the physical masses in practice. magnetic moments of the octet and decuplet baryons, −1 Standard values for the coupling constants are Fπ ¼ respectively. The results are: bM;D ≈ 0.321 GeV , bM;F ≈ 92.4 MeV, D ¼ 0.80, F ¼ 0.46. For the pion-nucleon 0.125 GeV−1, and d ≈−1.16 GeV−1. Furthermore, from 1 26 M coupling constant this implies gA ¼ F þ D ¼ . .In the radiative decay of a decuplet baryon to an octet baryon −1 the case of HA, we use estimates from large-Nc consid- and a , one obtains c ≈ 1.92 GeV . The values 9 ≈ 2 27 9 − 3 ≈ M erations: HA ¼ 5 gA . [15,23] or HA ¼ F D of the above NLO LECs come from [8]. Next we present 1.74 [14,26], since we lack a simple direct observable to pin two types of decay channels that can be used to determine ≈ 2 0 − 0 − it down. Numerically we use HA . . cE and the sign of cM, starting with Ω → Ξ e ν¯e. At NLO, we have five terms that contribute to the decays In view of the absence of data that one can use to help pin of interest [8]: two octet sector terms that are given by down cE and the sign of cM; we stress that the purpose of this paper is to motivate such experiments and not to ¯ μν σ ¯ μν σ bM;DtrðBffþ ; μνBgÞ þ bM;FtrðB½fþ ; μνBÞ; ð19Þ explore the uncertainties of already determined LECs. Therefore, we stick to the central values of those LECs one term from the decuplet sector given by for numerical results.

¯ μν c abd dMiðTμÞ ðf Þ Tν ; ð20Þ − 0 − abc þ d III. DECAY PROCESS Ω → Ξ e ν¯e and finally two decuplet-to-octet transition terms given by There are two contributing tree-level diagrams at NLO in χPT for the decay of the Omega baryon decaying into a ϵ ¯ eγ γ μν d abc icM adeBc μ 5ðfþ ÞbTν cascade baryon, an electron, and an electron antineutrino. ϵ ¯ eγ μν d abc These are shown in Fig. 1. þ icE adeBc μðf− ÞbTν þ H:c: ð21Þ At LO, we have three contributing terms: the ∼hA term μν from (6), the kinetic term for the mesons in (7), as well as The field strengths f are given by the standard weak charged-current interaction term [20]. μν μν † † μν Going to NLO, we also have the ∼c terms in (21). f ≔ uF u u F u ð22Þ M=E L R When calculating the decay width we used Mathematica FeynCalc with and to perform the traces of the gamma matrices [28,29]. The resulting partial decay width at μν ≔ ∂μ ν ν − ∂ν μ μ − μ μ ν ν NLO contains terms proportional to h2 , c2 , c2 and FR;L ðv a Þ ðv a Þ i½v a ;v a : A E M hAcE. Figure 2 illustrates the calculated and measured ð23Þ branching ratio. We use the values of the LECs described in Sec. II together with hA ¼ 2.0, i.e., the value of hA Interactions with external forces are studied by the replace- Ξ → Ξπ ν ν obtained from fitting the partial decay width of to ment of the vector and axial-vector sources v , a .For data, again see Table I. electromagnetic interactions we have the replacement [20], We then fit the NLO branching ratio to the measurement, 0 1 resulting in c : ð0.5 1Þ GeV−1 and ð−5 1Þ GeV−1. 2 00 E B 3 C The error comes from the experimental uncertainty of the B 0 − 1 0 C vμ → eAμ@ 3 A ð24Þ branching ratio. We cannot distinguish between the two 1 solutions of cE by only considering the integrated decay 00− 3 width, and likewise, we cannot pin down the sign of cM since the partial decay width contains no linear cM term. with the photon field Aμ and the e; and for weak interactions mediated by the W-bosons we have the replacement [20], 0 1 0 Vud Vus B C gw þ vμ − aμ → − pffiffiffi Wμ @ 00 0A þ H:c: ð25Þ 2 00 0

þ with the W-boson field Wμ , the Cabibbo-Kobayashi- − 0 − FIG. 1. Diagrams contributing to the decay of Ω → Ξ e ν¯e. Maskawa matrix elements Vud, Vus [27], and the weak These two diagrams are the only two topologically distinct gauge coupling gw (related to Fermi’s constant and the diagrams at NLO, but note that the ΩΞW-vertex comes from W mass). three terms.

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finds the majority of events in the Dalitz plots [i.e., closer to cosðθÞ¼1 or cosðθÞ¼−1]. Note also that there is an apparent antisymmetry such that cM → −cM is equivalent to cosðθÞ → − cosðθÞ. This is, however, only approxi- mately true when the masses are small compared to the masses. In the case of vanishing lepton masses, we find that all linear terms of cosðθÞ in the double differential decay width are proportional to cM. This phenomenon is called forward-backward asymmetry [30].

IV. DECAY PROCESSES B(J =3=2) → B(J =1=2)γπ

FIG. 2. NLO branching ratio of Ω− → Ξ0e−ν¯ as a function of For the decay of a decuplet baryon into an octet baryon, a e χ cE, together with the LO result and the experimental value [7]. pion and a photon we have six diagrams at NLO in PT. The gray box illustrates the measurement uncertainty. These diagrams are shown in Fig. 4. The interaction terms are given by all terms in (6), (7), (19), (20), and (21). We can instead study the distribution of the double As before, we use Mathematica and FeynCalc to perform differential decay width to obtain more information. In the traces of the gamma matrices. Furthermore, we explic- Fig. 3 we illustrate the double differential decay dis- itly checked that the Ward identity for the electromagnetic M μ 0 tribution for the four different solutions of cM and cE in current holds, that is μk ¼ for all decays. the frame where the electron and antineutrino goes back Concerning the values of the different LECs, we used ⃗ ⃗ 2 4 to back, i.e., k1 þ k2 ¼ 0. We chose to work with the hA ¼ . , being the average value of Table I, together with 2 − c 1.92 GeV−1 and c 0.5 GeV−1. The decays kinematic variables m ðe ν¯eÞ and cosðθÞ, where θ is the M ¼þ E ¼ angle between the three-momenta of the cascade baryon contain infrared divergences resulting from final-state and the electron. With enough statistics, it would be radiation that, in the limit of vanishing photon energy, possible to distinguish between all four cases, and even can make a propagating particle on shell [22]. Therefore, with less statistics, it could be possible to at least determine we used a cutoff of the photon energy at 25 MeV (in the the relative sign of cM and cE, depending on where one

0.06

0.04

0.04 0.03

0.02 0.02

0.01

0 0

(a) (b)

0.06

0.04

0.04 0.03

0.02 0.02

0.01

0 0

(c) (d)

FIG. 3. The double differential decay distribution of Ω− → 0 − Ξ ν¯ee for the two solutions of cE and for different signs of cM. The scale of the double differential decay distribution is linear FIG. 4. Diagrams contributing to the decay of a decuplet baryon with an arbitrary normalization. into an octet baryon, a pion and a photon.

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TABLE II. Branching ratios at LO and NLO of all energetically possible BðJ ¼ 3=2Þ → Bπγ decays allowed by flavor symmetry, except Σ → Λπγ that run over the pole of Σ0. The second column is showing the LEC dependence at NLO.

Decay LEC dependence at NLO BR at LO BR at NLO − − 0 −6 −6 Ξ → Ξ π γ hA;dM;bM;D;bM;F 8.2 × 10 8.6 × 10 − 0 − −3 −3 Ξ → Ξ π γ hA;HA;cM;cE;dM;bM;D 1.4 × 10 1.4 × 10 0 − þ −3 −3 Ξ → Ξ π γ hA;D;F;cM;cE;bM;D;bM;F 1.2 × 10 1.2 × 10 0 0 0 −6 Ξ → Ξ π γ hA;HA;D;F;cM;bM;D 0 1.9 × 10 þ þ 0 −7 −6 Σ → Σ π γ hA;HA;D;cM;dM;bM;D;bM;F 8.9 × 10 1.2 × 10 þ 0 þ −5 −5 Σ → Σ π γ hA;HA;F;cM;cE;dM;bM;D 3.6 × 10 3.7 × 10 − − 0 −7 −7 Σ → Σ π γ hA;dM;bM;D;bM;F 6.3 × 10 6.6 × 10 − 0 − −5 −5 Σ → Σ π γ hA;HA;cM;cE;dM;bM;D 4.4 × 10 4.5 × 10 0 þ − −5 −5 Σ → Σ π γ hA;HA;D;F;cM;cE;bM;D;bM;F 5.9 × 10 5.9 × 10 0 − þ −5 −5 Σ → Σ π γ hA;D;F;cM;cE;bM;D;bM;F 3.3 × 10 3.4 × 10 0 0 0 −8 Σ → Σ π γ hA;D;cM;bM;D 0 2.6 × 10 0 0 −6 Σ → Λπ γ hA;D;cM;bM;D 0 3.6 × 10 þþ þ −3 −3 Δ → pπ γ hA;HA;cM;cE;dM;bM;D;bM;F 1.7 × 10 1.8 × 10 þ 0 −5 −5 Δ → pπ γ hA;HA;D;F;cM;dM;bM;D;bM;F 5.6 × 10 7.2 × 10 þ þ −4 −4 Δ → nπ γ hA;HA;D;F;cM;cE;dM;bM;D 7.5 × 10 7.6 × 10 0 − −3 −3 Δ → pπ γ hA;HA;D;F;cM;dM;bM;D;bM;F 1.0 × 10 1.0 × 10 0 0 −6 Δ → nπ γ hA;HA;D;F;cM;bM;D 0 7.6 × 10 − − −3 −3 Δ → nπ γ hA;HA;cM;cE;dM;bM;D 2.3 × 10 2.3 × 10 frame where p⃗þ q⃗¼ 0), roughly corresponding to the much less) in the case of decays involving charged states. lowest detectable photon energy at the upcoming PANDA¯ The four decays with only neutral states (and vanishing experiment ( ANnihilation in DArmstadt) [31–33] decay width at LO) changed significantly (up to 80%). But and the Beijing Spectrometer III [34].Thiscutoffcanbe since they vanish at LO they are prone to be more sensitive arbitrarily chosen to match the photon energy resolution to the values of the NLO LECs. of any experiment. In Table II, we have collected the Once data are available, we can use these radiative predictions of all energetically possible decays and their three-body decays to further investigate cE. For this branching ratios at LO and NLO as well as the LEC purpose, the decays Ξ− → Ξ0π−γ and Ξ0 → Ξ−πþγ are dependence. of most interest, since they both have a relatively large Looking at Table II we see that decay widths containing branching ratio of ∼10−3 and because of the small (total) only neutral states vanish at LO, which is only natural since decay width of cascade baryons (as compared to broad neutral do not interact with photons at LO. Con- tributions with LO LECs appear at NLO since the ampli- tudes contain structures which are proportional to products of LO and NLO LECs, e.g., diagram 3 gives terms like ∼hAbM;D, but with absent pure LO contributions. Further- more, it is reassuring that the branching ratios of these neutral decays are small at NLO since they vanish at LO; indicating that the NLO contribution is, in general, a small correction. The branching ratios of decays with neutral pions in the final state are small due to the same reason, that is, since the pion-pion-photon vertex forbids neutral pions, the other- wise large contribution of the pion propagator in diagram 2 disappear. Let us briefly investigate the possible ranges of the different LECs due to their uncertainty, starting with HA. 0 3 FIG. 5. Impact of the two solutions of cE on the single dif- Varying the value of HA by . changes the decay widths 0 − þ −1 ferential decay width of Ξ → Ξ π γ using cM ¼þ1.92 GeV . insignificantly (much less than 1%), except in the cases of We plot the single differential decay width down to a photon 0 0 0 0 0 Ξ → Ξ π γ and Δ → nπ γ which change by 10% and energy of 50 MeV (otherwise the c -dependence is hard to −1 E 2%, respectively. We also considered cM ¼ −1.92 GeV , display). The gray intervals come from the uncertainty of the two which changes the decay widths by a few percents (often solutions of cE.

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Delta baryons). We find that the decay widths of these two decuplet to an octet baryon, parametrized by cE. The most − 0 − cascade decays decrease by around 20% when changing to promising decays for this purpose are Ω → Ξ e ν¯e, − 0 − 0 − þ the negative cE solution. In Fig. 5 we consider how the Ξ → Ξ π γ, and Ξ → Ξ π γ. Moreover, by studying 0 − single differential decay width of Ξ → Ξ πþγ changes the double differentiable decay distribution of the Omega when varying cE. decay, one can determine the sign of cM. We hope that such ¯ We note that at low photon energies both solutions of cE experiments can be carried out at PANDA and in part at converge, in the region where the single differential decay BES III. width blows up, due to the infrared divergence. Instead, it is at higher photon energies, i.e., lower m2ðπþΞ−Þ¼ 2 ACKNOWLEDGMENTS MΞ − 2MΞEγ, where we can distinguish between the two possible cE. The authors thank Albrecht Gillitzer for valuable dis- To conclude, we provide two types of decay channels cussions and his interest in a future measurement of the that can be used to probe the axial-vector transition of a radiative three-body decays using the PANDA¯ experiment.

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