Strong CP violation in spin-1/2 singly charmed baryons

We report on the calculation of the CP-violating form factor F3 and the corresponding electric dipole moment for charmed baryons in the spin-1/2 sector generated by the QCD θ-term. We work in the framework of covariant baryon chiral perturbation theory within the extended-on-mass-shell renormalization scheme up to next-to-leading order in the chiral expansion.


Introduction
Charge conjugation and parity symmetry (CP) violation is an essential condition for an asymmetry between matter and anti-matter in the present universe. On the other hand, CP violation by the complex phase of the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix is insufficient to explain the dominance of matter over antimatter [1,2], meaning that the presence of other CP violating mechanisms within or beyond the Standard Model (SM) is required. The QCD θ-term is the only source of P and T violation within the SM beyond the complex phase of the CKM quark-mixing matrix. However, because of the significant suppression of electric dipole moments (EDMs) induced by the complex phase of the CKM matrix and unobservably small CKM backgrounds, any measurement of EDM of any quantum system would indicate of presence CP violation beyond the CKM mechanism in the SM. EDMs are important observables generated by the CP-violating effects. Thus, measurements of hadron EDMs lead to severe restrictions in the mechanism generating CP violation, as detailed e.g. in ref. [3].
CP violation has recently been established in the charm sector, more precisely in the meson decays D 0 → K − K + and D 0 → π − π + [4], and LHCb has also measured the difference of CP-asymmetry of the three-body singly Cabibbo-suppressed Λ + c decays [5]. There have also been quite a number of studies predicting CP asymmetries in charmed baryon decays, see e.g. [6] and references therein. It is therefore of interest to investigate other possible effects of CP violation in singly-charmed baryons. Indeed, a first measurement of CP violation in Ξ + c → pK − π + decays has been performed by LHCb [7]. However, these data are consistent with the hypothesis of no CP violation. On the other hand, another experimental study promises to search for direct CP violation by measuring the asymmetries of three different decay channels of the Λ + c baryon [8]. Here, we concentrate on the effects generated by the strong CP-violating θ-term of QCD, that also induces electric dipole moments in light baryons, as pioneered in JHEP01(2021)115 refs. [10,11]. The proper framework to address such questions is baryon chiral perturbation theory, see [12] for a review. In fact, the masses, axial charges, and electromagnetic decays of the charmed and bottomed baryons have already been calculated in the framework of the heavy baryon approach [13,14]. More recently, the magnetic moments of the spin-1/2 singly charmed baryons were analyzed in covariant baryon chiral perturbation theory [15]. In this paper, we extend these studies and work out the CP-violating effects induced by the QCD θ-term. While there is experimental activity to assess the effects of the θ-term in neutron and proton EDMs, singly-charmed baryons offer a completely new venue towards these elusive effects, with very different systematic uncertainties that hamper such measurements. How competitive these measurements will be would require a much more refined analysis as presented here. We note that recent progress towards the first measurement of the charm baryon dipole moments has been reported in ref. [9], thus our investigation is very timely. In the near future, further measurements with charmed hadrons, along with different theoretical improvements, would help to further elucidate the CP violation in the charm quark sector.
The manuscript is organized as follows. In section 2, we briefly discuss the underlying chiral Lagrangian. The CP-violating electromagnetic form factor of the singly-charmed baryons is worked out in section 3 followed by the display of our numerical results in section 4. Section 5 contains the summary and outlook. The appendices contain some technicalities as well as more detailed tables of results.

Chiral Lagrangian including CP-violating terms
The QCD Lagrangian of the strong interactions including the θ term reads where G µν a is the gluon field-strength tensor, g is the strong coupling constant and M is the quark mass matrix. Strong CP violation arising from the U(1) anomaly in QCD is specified via the vacuum angle θ. The measurable quantity is not θ but the combination θ 0 = θ + arg detM, (2.2) because of the anomaly. Here, to describe the phenomena related to the θ-term, we seek a description in a properly tailored effective field theory, see e.g. refs. [16,17] for the detailed construction of the corresponding effective Lagrangian to one loop accuracy. The Goldstone bosons together with the flavor singlet η 0 , resulting from the spontaneous symmetry breaking of U(3) R × U(3) L into U(1) V , are represented by the matrixvalued fieldŨ . Treating the vacuum angle θ(x) as an external field, it transforms as θ(x) → θ (x) = θ(x) − 2N f α under axial U(1) rotations, with N f the number of flavors, and α is the rotation angle. Following the spontaneous chiral symmetry breaking, under the axial U(1) transformation,Ũ changes but the combination ofθ 0 (x) = θ(x) − i ln detŨ (x) stays invariant. Using this invariant combination ofθ 0 (x), one can construct the most general mesonic chiral effective Lagrangian up-to-and-including second chiral order

JHEP01(2021)115
Note that . . . denotes the trace in flavor space andχ = 2B 0 M, with the light quark mass matrix M = diag(m u , m d , m s ). The covariant derivative ofŨ is given by where v µ and a µ are the conventional vector and axial-vector external sources. The V i coefficients in the Lagrangian (2.3) are functions ofθ 0 . One needs to determine the vacuum expectation value ofŨ in order to include non-trivial vacuum effects based on the angle θ 0 . Parameterizing the vacuum as the minimized potential energy V (U 0 ) can be determined using the notationθ 0 = θ 0 − q ϕ q . In this way, the Taylor expansions of the V i functions in terms ofθ 0 yield Note that while all other V i are even function ofθ 0 , V 3 is odd. To express the Lagrangian in terms of the angles ϕ q one then writes theŨ with the vacuum expectation where φ represents the Goldstone boson octet Thus, the chiral effective Lagrangian in terms of the Goldstone boson fields composed iñ U reads [18] To leading order, A and B are given as After vacuum alignment, the V i coefficients are now functions ofθ 0 + √ 6η 0 /F 0 . Further, the normalization of the kinetic terms in the Lagrangian (2.3) provides (2.10)

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In principle, the coupling of the η 0 singlet is different from F π because the subgroup U(3) V does not present a nonet symmetry. However, in the large N c -limit F 0 = F π . Moreover, the quantity ofθ 0 can be denoted in terms of physical quantities [21] θ 0 = 1 + 4V Here, we note thatθ 0 = O(δ 2 ), and take 1/N c = O(δ 2 ) as counting rules [19]. More detail and information on the formalism used in the work can be found in e.g. in refs. [18,20]. We now turn to the baryon sector of the effective Lagrangian. In the SU(3) flavor representation the spin-1/2 anti-symmetric triplet and symmetric sextet charmed baryon states are denoted as in the following matrices, respectively, Similarly to the mesonic Lagrangian one can write down the most general effective Lagrangian for the charmed baryon multiplets. Here, we only present the terms pertinent to the calculation. In the quark mass and momentum expansion, the relevant free and interaction Lagrangians up to the second chiral order are given by [13,15,18,22,24],

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where the relevant building blocks arẽ (2.13) The charge matrix for the singly-charmed baryons is Q h = diag (1, 0, 0), while for the light quarks the charge matrix is Q l = diag(2/3, −1/3, −1/3). We use w 10/11 + 3w 12 = w 10 as in ref. [21]. As can be seen from the contributing Lagrangians, there are quite number of low-energy constants (LECs). The meson-baryon coupling constants g i (i = 1, . . . , 6), the symmetrybreaking LECs b D and b F as well as the LECs w 16/17 , w 18 related to the CP-conserving electromagnetic response can all be taken from earlier studies of different observables, as detailed in section 4.
This leaves us with the yet undetermined LECs w 10 , w 13/14 , w 15 and w 13/14 , w 15 . As will be shown, we can fix w 13/14 , w 15 from recent lattice results QCD for the neutron and proton electric dipole moments, d n and d p , respectively. The remaining of these LECs will be varied as 0 +0.5 −0.5 GeV −1 , that is within a natural range. This naive dimensional analysis should be eventually overcome by a more sophisticated modeling of the LECs or invoking further lattice QCD results. Having fixed/estimated all the LECs will then allow to estimate the CP-violating contributions to the singly-charmed baryons induced by the θ-term.

CP-violating electromagnetic form factor
The electromagnetic form factors of a baryon are defined via the matrix element of the electromagnetic current, with q 2 = (p f − p i ) 2 the invariant momentum transfer squared, m B the baryon mass and J µ em the electromagnetic current. Here, F 1 (q 2 ) and F 2 (q 2 ) are the P-and CP-conserving Dirac and Pauli form factors, respectively. F A (q 2 ) denotes the P-violating anapole form factor, and F 3 (q 2 ), which will be considered throughout this work, the P-and CP-violating electric dipole form factor. The electric dipole moment of the baryon B is then given by In what follows, we will use the effective Lagrangian to calculate the CP-violating form factor of the singly-charmed baryons at next-to-leading (NLO) order, which includes tree as well as loop diagrams as shown in figure 1, where we display the corresponding JHEP01(2021)115 Feynman diagrams. Tree-level diagrams at leading order are presented in (a) and (b). One-loop diagrams at order O(δ 2 ) and O(δ 3 ) in (c)-(d), and (e)-(h), respectively. The type of diagrams in (g)-(h) with pionic or kaonic loops of the antitriplet and the sextet charmed baryons are canceling each other, thus they are not displayed here.
We show different combinations of the charmed baryon states from anti-triplet and sextet multiplets considered throughout the calculation in figure 2.
The results obtained for the form factor F 3 (q 2 ) of the charmed baryons coming from the tree-level diagrams are collected in table 1 with α = 576V As usual in the EOMS scheme, the loop contributions are rather lengthy expression. Let us discuss the case of the Λ + c . The one-loop contribution can be written as, cf. figure 1,  Diagram type number meson-baryon state Coefficient Table 2. Loop contribution to the F 3 (q 2 ) of the Λ + c baryon with β = (b D/F + b 0 + 3w 10 ).

JHEP01(2021)115 4 Results
First, we must fix parameters. The pion decay constant is taken as F π = 92.2 MeV. In what follows, due to the lack of data from the charmed meson sector, we make recourse to the ground state baryon octet as much as possible to fix as many LECs as possible. While this is an approximation, we expect that we are estimating at least the right order of magnitude of the EDMs of the charmed baryons. Consequently, two symmetry-breaking LECs in the baryon sector can be obtained from baryon mass splittings. We use b D = −0.606 GeV −1 and b F = −0.209 GeV −1 [23,24]. The tree-level contributions can be expressed in terms of two independent linear combinations of unknown LECs as α(w 13 [27]. The various baryon-meson couplings are taken from refs. [13,14], g 1 = 0.98, g 2 = −0.60, g 3 = 0.85, and g 4 = 1.04. Because of the forbidden B3B3φ-vertex, we have g 6 = 0. We use the physical masses of the pertinent mesons and baryons running in the corresponding loops, cf. tables 4-11. As the unknown LECs cannot be parameterized such a common constant as in [24], since the combinations coming from different particles are different, they have to be considered individually. Using the lattice data from [28] at physical pion mass, we use the neutron dipole moment to fix βw 15 from the Ξ 0 c by comparing the loop contributions. With that, we can use the proton electric dipole moment to determine βw 13/14 from the Λ + c . We get βw 13 With these obtained values, we take the variation of w 10 , w 13/14 and w 15 , and calculate the CP-violating form factor F 3 (q 2 ) for the singly-charmed baryons in the range q 2 0.05 . . . 0.3 GeV 2 as given in tables 12-14. We are well aware that there are other determinations of d p and d n in the literature, see e.g. refs. [29,30], and that there is an on-going debate on the axial rotation in the finite volume (mixing between the form factors F 2 and F 3 , see e.g. ref. [31]). However, since our study is largely exploratory, we do not explore the whole possible parameter space.
The electric dipole moments for the various baryons are collected in table 3. As there is a sizeable uncertainty induced by the unknown LECs, we refrain from performing a systematic error analysis accounting e.g. for the effects of higher orders in the chiral expansion. Hopefully, lattice QCD will be able to supply pertinent information on the LECs so that more accurate predictions can be made.

Conclusion
In this paper, we have performed a one-loop calculation of the CP-violating form factor F 3 (q 2 ) and the corresponding electric dipole moments of the spin-1/2 singly-charmed  Table 3. Electric dipole moments for the singly-charmed baryons in units of e θ 0 fm. Set 1,2,3 refers to w 10 = w 13/14 = w 15 = −0.5, 0, +0.5, in order. baryons, where the mechanism of the CP violation is the QCD θ-term. Not all the appearing low-energy constants could be fixed from experimental or lattice QCD data, so the resulting predictions show a spread, cf. table 3 and the tables in appendix C. We hope that with more lattice QCD studies on strong CP violations, these LECs can be determined and more accurate predictions can be made, not to mention possible experimental determinations.