Magnetic moments of the spin-1/2 singly charmed baryons in covariant baryon chiral perturbation theory
MMagnetic moments of the spin-1 / Rui-Xiang Shi, Yang Xiao, and Li-Sheng Geng ∗ School of Physics and Nuclear Energy Engineering & International Research Center for Nuclei andParticles in the Cosmos & Beijing Key Laboratory of Advanced Nuclear Materials and Physics,Beihang University, Beijing 100191, China
Abstract
Recent experimental advances have reignited theoretical interests in heavy-flavor hadrons. In this work, we studythe magnetic moments of the spin-1 / g − are fixed with the help of the quark model and the heavy quark spin flavor symmetry, while the remaining d , d , d and d are determined by fitting to the lattice QCD pion-mass dependent data. We study the magnetic momentsas a function of m π and compare our results with those obtained in the heavy baryon chiral perturbation theory. Wefind that the loop corrections induced by the anti-triplet states are dominated by the baryon pole diagram. In addi-tion, we predict the magnetic moments of the spin-1 / PACS numbers: ∗ E-mail: [email protected] a r X i v : . [ h e p - ph ] D ec . INTRODUCTION In the last two decades, tremendous progress has been made in our understanding of heavy-flavorhadrons, thanks to the experimental discoveries by collaborations such as LHCb, BELLE, and BESIII andthe related theoretical studies. In the charmed baryon sector, 24 singly charmed baryons and two doublycharmed baryons are listed in the current version of the review of particle physics [1]. Among them, thenewest members include the Λ c (2860) [2], the five Ω c states [3], and the Ξ ++ cc [4]. Inspired by these andother experimental discoveries, there are extensive theoretical and lattice QCD studies on their nature andtheir decay and production mechanisms (see, e.g., Refs. [5–12] references cited therein).The magnetic moment of a baryon plays an extremely important role in understanding its internal struc-ture. Historically, the experimental measurement of the magnetic moments of the proton and the neutronrevealed that they are not point-like particles. The subsequent studies helped the establishment of the quarkmodel as well the theory of the strong interaction, Quantum Chromo Dynamics. Unlike those of the ground-state baryons, the magnetic moments of the spin-1 / / ff ective field theory of QCD, is an appropriateframework to study the magnetic moments of hadrons, particularly, their light quark mass dependence. Itprovides a systematic expansion of physical observables in powers of ( p / Λ χ ) n χ , where p is a small mo-mentum and Λ χ is the chiral symmetry breaking scale. However, its application to the one-baryon sectorencountered a di ffi culty, i.e., a systematic power counting (PC) is lost due to the large non-vanishing baryonmass m in the chiral limit. Over the years, three approaches were proposed to overcome this issue, i.e., theHB [31, 32], the infrared (IR) [33], and the EOMS [34] schemes. The IR and the EOMS schemes are the2elativistic formulations of BChPT. A brief summary and comparison of the three di ff erent approaches canbe found in Ref. [35].The EOMS scheme is di ff erent from the HBChPT, because it retains a series of higher-order termswithin the covariant power counting (PC) rule when removing the power-counting-breaking (PCB) terms. Inrecent years, many physical observables have been successfully studied in this scheme such as the magneticmoments [29, 36–40], the masses and sigma terms [28, 41–43] of the octet, decuplet and spin-1 / / ff ective Lagrangians and calculate therelevant Feynman diagrams up to O ( p ). Results and discussions are given in Sec. III, followed by a shortsummary in Sec. IV. II. THEORETICAL FORMALISM
The magnetic moments of singly charmed baryons are defined via the matrix elements of the electro-magnetic current J µ as follows: (cid:104) ψ ( p f ) | J µ | ψ ( p i ) (cid:105) = ¯ u ( p f ) (cid:34) γ µ F B ( q ) + i σ µν q ν m B F B ( q ) (cid:35) u ( p i ) , where ¯ u ( p f ) and u ( p i ) are the Dirac spinors, m B is the singly charmed baryon mass, and F B ( q ) and F B ( q )denote the Dirac and Pauli form factors, respectively. The four-momentum transfer is defined as q = p i − p f . At q = F B (0) is the so-called anomalous magnetic moment, κ B , and the magnetic moment is µ B = κ B + Q B , where Q B is the charge of the singly charmed baryon.The five Feynman diagrams contributing to µ B up to O ( p ) are shown in Fig.1. The leading ordercontribution of O ( p ) is provided by the following Lagrangian: L (2)33 = d m ¯3 Tr( ¯ B ¯3 σ µν F + µν B ¯3 ) + d m ¯3 Tr( ¯ B ¯3 σ µν B ¯3 )Tr( F + µν ) , L (2)66 = d m Tr( ¯ B σ µν F + µν B ) + d m Tr( ¯ B σ µν B )Tr( F + µν ) , (1)where the numbers in the superscript are the chiral order, σ µν = i [ γ µ , γ ν ], F + µν = | e | ( u † Q h F µν u + uQ h F µν u † ), F µν = ∂ µ A ν − ∂ ν A µ , and Q h = diag(1 , ,
0) is the charge operator of the charmed baryon, u = exp[ i Φ / F φ ],3 d) FIG. 1: Feynman diagrams contributing to the singly charmed baryon magnetic moments up to NLO. Diagram (a)contributes at LO, while the other diagrams contribute at NLO. The solid, dashed, and wiggly lines represent singlycharmed baryon, Goldstone bosons, and photons, respectively. The heavy dots denote the O ( p ) vertices. with the unimodular matrix containing the pseudoscalar nonet, and F φ the pseudoscalar decay constant.In the following analysis, we take F π = . F K = . F π , and F η = . F π . In the SU(3) flavorrepresentation, there are three kinds of singly charmed baryons, which are denoted as B ¯3 , B , and B ∗ µ ,respectively, B ¯3 = Λ + c Ξ + c − Λ + c Ξ c − Ξ + c − Ξ c , B = Σ ++ c Σ + c √ Ξ (cid:48) + c √ Σ + c √ Σ c Ξ (cid:48) c √ Ξ (cid:48) + c √ Ξ (cid:48) c √ Ω c , B ∗ µ = Σ ∗ ++ c Σ ∗ + c √ Ξ ∗ + c √ Σ ∗ + c √ Σ ∗ c Ξ ∗ c √ Ξ ∗ + c √ Ξ ∗ c √ Ω ∗ c . (2)The spin of the B ¯3 and B states is 1 / B ∗ µ states is 3 / m ¯3 = m = m ∗ = ff erences are δ = m − m ¯3 =
127 MeV, δ = m ∗ − m ¯3 =
194 MeV, and δ = m ∗ − m =
67 MeV.The loop diagrams arising at NLO are determined in terms of the lowest order LECs from4 (1) B + L (1) MB + L (2) M , which are, L (1) B =
12 Tr[ ¯ B ¯3 ( i / D − m ¯3 ) B ¯3 ] + Tr[ ¯ B ( i / D − m ) B ] + Tr[ ¯ B ∗ µ ( − g µν ( i / D − m ∗ ) + i ( γ µ D ν + γ ν D µ ) − γ µ ( i / D + m ∗ ) γ ν B ∗ ν ] , L (1) MB = g B / u γ B ] + g B / u γ B ¯3 + h . c . ] + g B ∗ µ u µ B + h . c . ] + g B ∗ µ u µ B ¯3 + h . c . ] + g B ∗ µ / u γ B ∗ µ ] + g B ¯3 / u γ B ¯3 ] , L (2) M = F φ ∇ µ U ( ∇ µ U ) † ] , (3)with D µ B = ∂ µ B + Γ µ B + B Γ T µ , Γ µ =
12 ( u † ∂ µ u + u ∂ µ u † ) − i u † v µ u + uv µ u † ) = − ieQ h A µ , u µ = i ( u † ∂ µ u − u ∂ µ u † ) + ( u † v µ u − uv ν u † ) , U = u = e i Φ F φ , ∇ µ U = ∂ µ U + ieA µ [ Q l , U ] , (4)where v µ stands for the vector source, and the charge matrix for the light quark is Q l = diag(2 / , − / , − / B ¯3 state. Considering parityand angular momentum conservation, the B ¯3 B ¯3 φ vertex is forbidden, i.e., g = B ¯3 and B states, the tree level contributions of the magnetic moments can be easily obtainedfrom Eq. (1), which are: κ ( a , = α ¯3 d + β ¯3 d ,κ ( a , = α d + β d . (5)The values of α ¯3 , β ¯3 , α , and β are tabulated in Table I and Table II. The four LECs d , d , d and d willbe determined by fitting to lattice QCD data.At O ( p ), the loop contributions to the magnetic moments, which come from diagrams (b), (c), (d), and5 ABLE I: Coe ffi cients of the tree level contributions of Eq. (5) for the B ¯3 states. Λ + c Ξ + c Ξ c α ¯3 12 12 β ¯3 (e) in Fig. 1, are written as, κ (3)¯3 = π (cid:88) φ = π, K g F φ ξ (3 , b ) B ¯3 φ,δ H ( b ) B ¯3 ( δ , m φ ) + (cid:88) φ = π, K g F φ ξ (3 , c ) B ¯3 φ,δ H ( c ) B ¯3 ( δ , m φ ) + (cid:88) φ = π, K ,η g F φ ξ (3 , d ) B ¯3 φ,δ H ( d ) B ¯3 ( δ , m φ ) + (cid:88) φ = π, K ,η g F φ ξ (3 , e ) B ¯3 φ,δ H ( e ) B ¯3 ( δ , m φ ) ,κ (3)6 = π (cid:88) φ = π, K g F φ ξ (3 , b ) B φ H ( b ) B (0 , m φ ) + (cid:88) φ = π, K g F φ ξ (3 , b ) B φ,δ H ( b ) B ( δ , m φ ) + (cid:88) φ = π, K g F φ ξ (3 , c ) B φ,δ H ( c ) B ( δ , m φ ) + (cid:88) φ = π, K ,η g F φ ξ (3 , d ) B φ H ( d ) B (0 , m φ ) + (cid:88) φ = π, K ,η g F φ ξ (3 , d ) B φ,δ H ( d ) B ( δ , m φ ) + (cid:88) φ = π, K ,η g F φ ξ (3 , e ) B φ,δ H ( e ) B ( δ , m φ ) , (6)with the coe ffi cients ξ (3; b , c , d , e ) B ¯3 φ,δ i , ξ (3; b , c , d , e ) B φ,δ i listed in Table III and Table IV. The explicit expressions of the loopfunctions H ( b , c , d , e ) B ¯3 ( δ i , m φ ) and H ( b , c , d , e ) B ( δ i , m φ ) can be found in the Appendix.Once we obtain the loop functions in the EOMS scheme, we can easily obtain their HB counterpartsby performing 1 / m expansions, We have checked that our results agree with those of Ref. [23]. In thefollowing section, for the sake of comparison, we study also the performance of the HBChPT in describingthe lattice QCD data of Refs. [24–26]. It should be noted that in the following section, unless otherwise6tated, the HBChPT results refer to the ones obtained in the present work, not those of Ref. [23] TABLE II: Coe ffi cients of the tree level contributions of Eq. (5) for the B states. Σ ++ c Σ + c Σ c Ξ (cid:48) + c Ξ (cid:48) c Ω c α β ffi cients of the loop contributions of Eq. (6) for the B ¯3 states. Λ + c Ξ + c Ξ c ξ (3 , b ) B ¯3 π,δ − ξ (3 , b ) B ¯3 K ,δ − ξ (3 , c ) B ¯3 π,δ − ξ (3 , c ) B ¯3 K ,δ − ξ (3 , d ) B ¯3 π,δ ξ (3 , d ) B ¯3 K ,δ ξ (3 , d ) B ¯3 η,δ ξ (3 , e ) B ¯3 π,δ
14 12 ξ (3 , e ) B ¯3 K ,δ
12 52 12 ξ (3 , e ) B ¯3 η,δ III. RESULTS AND DISCUSSIONS
In this section, we determine the LECs d , d , d and d by fitting to the lattice QCD data of Refs. [24–26], which are collected in Table V for the sake of easy reference. Because of the limited lattice QCD data,the other LECs g − are fixed by the quark model and the heavy quark spin flavor symmetry. Their valuesare g = . g = − (cid:113) g = − . g = √ g = .
85, and g = − √ g = .
04 [50, 50, 51]. In ourleast-squares fit, the χ as a function of the LECs is defined as χ (C X ) = n (cid:88) i = ( µ th i (C X ) − µ LQCD i ) σ i , (7)where C X denote all the LECs, σ i correspond to the uncertainty of each lattice QCD datum, µ th i (C X ) and µ LQCD i stand for the magnetic moments obtained in the BChPT and those of the lattice QCD in Table V,respectively. 7 ABLE IV: Coe ffi cients of the loop contributions of Eq. (6) for the B states. Σ ++ c Σ + c Σ c Ξ (cid:48) + c Ξ (cid:48) c Ω c ξ (3 , b ) B π − − ξ (3 , b ) B K − -1 ξ (3 , b ) B π,δ − − ξ (3 , b ) B K ,δ − − ξ (3 , c ) B π,δ − − ξ (3 , c ) B K ,δ − -1 ξ (3 , d ) B π
14 12 ξ (3 , d ) B K
52 12 ξ (3 , d ) B η
23 13 ξ (3 , d ) B π,δ ξ (3 , d ) B K ,δ ξ (3 , d ) B η,δ ξ (3 , e ) B π,δ
12 18 14 ξ (3 , e ) B K ,δ
12 14
54 14 12 ξ (3 , e ) B η,δ
13 16 ff erent m π [24–27], in units of nuclear magneton [ µ N ]. m π (MeV) Ξ + c Ξ c Σ ++ c Σ c Ξ (cid:48) + c Ξ (cid:48) c Ω c Phys. · · · · · · . − . · · · · · · − . . . · · · · · · . − . − . · · · · · · . − . · · · · · · − . · · · · · · . − . · · · · · · − . · · · · · · . − . · · · · · · − . · · · · · · . − . · · · · · · − . In order to decompose the contributions of loop diagrams, we will consider two cases. In case 1, all theallowed intermediate baryons are taken into account, while in case 2, only intermediate baryons of the sametype as those of the external baryons are considered. Fitting to the lattice QCD data of Table V and with g − fixed, the resulting LECs and χ are listed in Table VI. One notes that the EOMS BChPT descriptionsof the lattice QCD data are better than that of the HB BChPT in both cases.8 ABLE VI: LECs d , d , d , and d determined by fitting to the lattice QCD data, with g − fixed. In case 1 all theallowed intermediate baryons in the loop diagrams are taken into account, while in case 2 only intermediate baryonsof the same type as those of the external baryons in the loop diagrams are considered.Case 1 Case 2EOMS 1 HB 1 EOMS 2 HB 2 d − . − . − . − . d . . . . d . . . . d − . − . − . − . g .
98 0 .
98 0 .
98 0 . g − .
60 0 − .
60 0 g .
85 0 .
85 0 .
85 0 . g .
04 0 1 .
04 0 χ .
42 131 .
05 15 .
10 34 . Note that we do not fit to the lattice QCD data obtained at m π =
700 MeV, which are probably out ofthe range of validity of NLO ChPT. Furthermore, as can be seen in Fig. 3, the di ff erence between the latticeQCD value and the ChPT prediction for µ Ξ (cid:48) c is somehow relatively large. Thus, we do not include the latticeQCD magnetic moment of Ξ (cid:48) c in our fitting as well.For the sake of comparison with the lattice QCD data, in Fig. 2, we present the predicted magneticmoments of the singly charmed anti-triplet baryons as a function of m π . It is seen that the EOMS BChPTresults are of the same qualify as those of the HB BChPT for Ξ + c and Ξ c . However, surprisingly, the EOMSand HB predictions for Λ + c in case 1 are very di ff erent. From Table VII, we note that in the HBChPT thecontributions from the intermediate anti-triplet and sextet baryons cancel each other at O ( p ). Thus, at thisorder, loop corrections are quite small. But in the EOMS scheme, the loop contributions are rather large,especially for Λ + c . In addition, we note that the main contributions of the loop diagrams are from the baryonpole diagram. Therefore, the large di ff erence for the prediction of µ Λ + c is caused by the absence of thebaryon pole diagram in the HB BChPT at O ( p ).In Fig. 3, we plot the predicted magnetic moments of the singly charmed sextet baryons as a function of m π , in comparison with the lattice QCD data. The EOMS BChPT results are in better agreement with thelattice QCD data than those of the HB BChPT. As shown in Tables VI, on average the description of thelattice QCD data becomes worse if the intermediate anti-triplet states are included. Therefore, on averagethe results obtained in case 2 are in better agreement with the lattice QCD data. This has been noted in9ef. [23] as well. In Tables VII and VIII, we decompose the loop contributions mediated by the ¯3, 6, and6 ∗ states. One can see that the convergence pattern in case 2 is in generally better than that in case 1, withprobably the exception of Σ c . Therefore, we take the predictions obtained in case 2 as our final results.In Fig. 4 and Fig. 5, we compare the predicted magnetic moments of all the singly charmed baryons atthe physical point with those obtained in other approaches. We note that the results of di ff erent approachesare rather scattered. However, our results are in better agreement with those of the HBChPT of Ref. [23],though we have chosen di ff erent strategies to determine some of the LECs. Clearly, further experimental orlattice QCD studies are needed to pin down their values and to discriminate between di ff erent theoreticalapproaches. -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0-0 .8-0 .6-0 .4-0 .20 .00 .20 .40 .60 .81 .01 .2 mX mN ) m p ( G e V ) E O M S 1 H B 1 E O M S ( H B ) 2 mX +c( mN ) m p ( G e V ) E O M S 1 H B 1 E O M S ( H B ) 2 mL +c( mN ) m p ( G e V ) E O M S 1 H B 1 E O M S ( H B ) 2
L Q C D N o n - p h y s .
L Q C D N o n - p h y s .
FIG. 2: Magnetic moments of the singly charmed anti-triplet baryons as a function of m π . The solid black nablasrepresent the corresponding lattice QCD data that are fitted.TABLE VII: Decomposition of the loop contributions to the magnetic moments of singly charmed baryons. Thesubscript ¯3, 6, and 6 ∗ denote the loop diagrams with the intermediate ¯3, 6, and 6 ∗ states at O ( p ), respectively.EOMS 1 HB 1 LQCD [24,27] O ( p ) O ( p ) ¯3 O ( p ) O ( p ) ∗ µ tot O ( p ) O ( p ) ¯3 O ( p ) O ( p ) ∗ µ tot B ¯3 µ Λ + c . · · · . − . − .
232 0 . · · · − .
263 0 .
280 0 . · · · µ Ξ + c . · · · . − .
913 0 .
233 0 . · · · − .
169 0 .
215 0 . · · · µ Ξ c . · · · . − .
996 0 .
193 0 . · · · . − .
495 0 . · · · B µ Σ ++ c . − . − .
444 0 .
090 1 .
604 3 . − . − .
988 0 .
288 2 .
897 1 . µ Σ + c . − . − . − .
042 0 .
100 1 . − . − .
349 0 .
091 0 . · · · µ Σ c − . − .
090 0 . − . − . − . − .
168 0 . − . − . − . µ Ξ (cid:48) + c .
428 0 .
067 0 . − .
048 0 .
559 1 .
044 0 . − .
145 0 .
053 1 . · · · µ Ξ (cid:48) c − .
394 0 .
135 0 . − . − . − .
828 0 .
159 0 . − . − . · · · µ Ω c − .
394 0 .
361 0 . − . − . − .
828 0 .
486 0 . − . − . − . . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 . 3 50 . 00 . 51 . 01 . 52 . 02 . 53 . 03 . 54 . 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 . 3 50 . 00 . 20 . 40 . 60 . 81 . 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 . 3 5- 2 . 5- 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 00 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 . 3 50 . 00 . 20 . 40 . 60 . 81 . 01 . 21 . 4 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 . 3 5- 1 . 6- 1 . 2- 0 . 8- 0 . 40 . 00 . 4 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 . 3 5- 1 . 2- 1 . 0- 0 . 8- 0 . 6- 0 . 4- 0 . 20 . 0 mS ++c( mN ) m p ( G e V ) E O M S 1 E O M S 2 H B 1 H B 2 mS +c( mN ) m p ( G e V ) E O M S 1 E O M S 2 H B 1 H B 2 mS mN ) m p ( G e V ) E O M S 1 E O M S 2 H B 1 H B 2 mX ’+c( mN ) m p ( G e V ) E O M S 1 E O M S 2 H B 1 H B 2 mX ’0c( mN ) m p ( G e V ) E O M S 1 E O M S 2 H B 1 H B 2 mW mN ) m p ( G e V ) E O M S 1 E O M S 2 H B 1 H B 2 L Q C D N o n - p h y s .
L Q C D N o n - p h y s .
L Q C D N o n - p h y s .
L Q C D P h y s . L Q C D P h y s .
L Q C D P h y s .
L Q C D N o n - p h y s .
L Q C D N o n - p h y s .
FIG. 3: Magnetic moments of the singly charmed sextet baryons as a function of m π . The solid black nablas referto the corresponding lattice QCD data fitted. The hollow nablas stand for the lattice QCD physical values. The bluenablas denote the lattice QCD data not used in our fitting.TABLE VIII: Same as Table VII , but for case 2.EOMS 2 HB 2 LQCD [24,27] O ( p ) O ( p ) ¯3 O ( p ) O ( p ) ∗ µ tot O ( p ) O ( p ) ¯3 O ( p ) O ( p ) ∗ µ tot B ¯3 µ Λ + c . · · · · · · · · · .
235 0 . · · · · · · · · · . · · · µ Ξ + c . · · · · · · · · · .
235 0 . · · · · · · · · · . · · · µ Ξ c . · · · · · · · · · .
192 0 . · · · · · · · · · . · · · B µ Σ ++ c . · · · − .
444 0 .
090 1 .
285 2 . · · · − .
988 0 .
288 2 .
003 1 . µ Σ + c . · · · − . − .
042 0 .
190 0 . · · · − .
349 0 .
091 0 . · · · µ Σ c − . · · · . − . − . − . · · · . − . − . − . µ Ξ (cid:48) + c . · · · . − .
048 0 .
390 0 . · · · − .
145 0 .
053 0 . · · · µ Ξ (cid:48) c − . · · · . − . − . − . · · · . − . − . · · · µ Ω c − . · · · . − . − . − . · · · . − . − . − . IV. SUMMARY
Motivated by the recent experimental progress on heavy flavor hadrons, we have studied the magneticmoments of the singly charmed baryons in the covariant baryon chiral perturbation theory (BChPT) up to11 . 2 0 . 3 0 . 4 0 . 50 . 2 0 . 3 0 . 4 0 . 50 . 2 0 . 3 0 . 4 0 . 5
H B C h P T , 1 8H B C h P T , 1 8H B C h P T , 1 8N . B a r i k e t a l . , 8 3B . J u l i a - D i a z e t a l . , 0 4S . K u m a r e t a l . , 0 5A . F a e s s l e r e t a l . , 0 6N . S h a r m a e t a l . , 1 0A . B e r n o t a s e t a l . , 1 2T h i s w o r k m X ( m N ) N . B a r i k e t a l . , 8 3B . J u l i a - D i a z e t a l . , 0 4S . K u m a r e t a l . , 0 5A . F a e s s l e r e t a l . , 0 6B . P a t e l e t a l . , 0 8N . S h a r m a e t a l . , 1 0A . B e r n o t a s e t a l . , 1 2T h i s w o r k m L +c ( m N ) N . B a r i k e t a l . , 8 3B . J u l i a - D i a z e t a l . , 0 4S . K u m a r e t a l . , 0 5A . F a e s s l e r e t a l . , 0 6N . S h a r m a e t a l . , 1 0A . B e r n o t a s e t a l . , 1 2T h i s w o r k m X +c ( m N ) FIG. 4: Magnetic moments of the anti-triplet baryons obtained in di ff erent approaches. The light-blue bands representthe result obtained in the present work. The others are taken from Ref. [13] (N. Barik et al., 83), Ref. [14] (B. Julia-Diaz et al., 04), Ref. [15] (S. Kumar et al., 05), Ref. [16] (A. Faessler et al., 06), Ref. [17] (B. Patel et al., 08), Ref. [18](N. Sharma et al., 10), Ref. [19] (A. Bernotas et al., 12), and Ref. [23] (HB ChPT, 18). . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7- 2 . 0 - 1 . 8 - 1 . 6 - 1 . 4 - 1 . 2 - 1 . 0 - 0 . 8 - 0 . 6 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2- 1 . 6 - 1 . 4 - 1 . 2 - 1 . 0 - 0 . 8 - 0 . 6 - 1 . 0 - 0 . 9 - 0 . 8 - 0 . 7 - 0 . 6 - 0 . 5 H B C h P T , 1 8 H B C h P T , 1 8H B C h P T , 1 8 H B C h P T , 1 8H B C h P T , 1 8 J . Y . K i m e t a l . , 1 8J . Y . K i m e t a l . , 1 8 J . Y . K i m e t a l . , 1 8J . Y . K i m e t a l . , 1 8J . Y . K i m e t a l . , 1 8N . B a r i k e t a l . , 8 3B . J u l i a - D i a z e t a l . , 0 4S . K u m a r e t a l . , 0 5A . F a e s s l e r e t a l . , 0 6B . P a t e l e t a l . , 0 8N . S h a r m a e t a l . , 1 0A . B e r n o t a s e t a l . , 1 2T h i s w o r k N . B a r i k e t a l . , 8 3B . J u l i a - D i a z e t a l . , 0 4S . K u m a r e t a l . , 0 5A . F a e s s l e r e t a l . , 0 6B . P a t e l e t a l . , 0 8N . S h a r m a e t a l . , 1 0A . B e r n o t a s e t a l . , 1 2G . - S Y a n g e t a l . , 1 8A . F a e s s l e r e t a l . , 0 6 A . F a e s s l e r e t a l . , 0 6B . P a t e l e t a l . , 0 8 B . P a t e l e t a l . , 0 8N . S h a r m a e t a l . , 1 0 N . S h a r m a e t a l . , 1 0A . B e r n o t a s e t a l . , 1 2 A . B e r n o t a s e t a l . , 1 2L Q C D , 1 4G . - S Y a n g e t a l . , 1 8 G . - S Y a n g e t a l . , 1 8T h i s w o r k T h i s w o r kN . B a r i k e t a l . , 8 3S . - L Z h u e t a l . , 9 7B . J u l i a - D i a z e t a l . , 0 4S . K u m a r e t a l . , 0 5A . F a e s s l e r e t a l . , 0 6B . P a t e l e t a l . , 0 8
G . - S Y a n g e t a l . , 1 8 m S + +c ( m N ) A . F a e s s l e r e t a l . , 0 6B . P a t e l e t a l . , 0 8N . S h a r m a e t a l . , 1 0A . B e r n o t a s e t a l . , 1 2L Q C D , 1 5G . - S Y a n g e t a l . , 1 8T h i s w o r kT h i s w o r kN . B a r i k e t a l . , 8 3 N . B a r i k e t a l . , 8 3S . - L Z h u e t a l . , 9 7B . J u l i a - D i a z e t a l . , 0 4 B . J u l i a - D i a z e t a l . , 0 4S . K u m a r e t a l . , 0 5 S . K u m a r e t a l . , 0 5 m S +c ( m N ) J . Y . K i m e t a l . , 1 8 m S ( m N ) m X ’ +c ( m N ) N . S h a r m a e t a l . , 1 0A . B e r n o t a s e t a l . , 1 2L Q C D , 1 4G . - S Y a n g e t a l . , 1 8 m X ’ 0c ( m N ) N . B a r i k e t a l . , 8 3B . J u l i a - D i a z e t a l . , 0 4S . K u m a r e t a l . , 0 5
H B C h P T , 1 8
T h i s w o r k m W ( m N ) FIG. 5: Same as Fig. 4, but for the sexet baryons. Additional data are taken from Ref. [20] (S.-L Zhu et al., 97),Ref. [21] (G.-S Yang et al., 18), Ref. [22] (J. Y. Kim et al., 18), Ref. [24] (LQCD, 14), and Ref. [27] (LQCD, 15). the next-to-leading order. Using the quark model and the heavy quark spin flavor symmetry to fix some of13he low energy constants, we determined the rest by fitting to the lattice QCD data. We compared our resultswith those of the heavy baryon (HB) ChPT and found that on average the lattice QCD quark mass dependentdata can be better described by the covariant BChPT, consistent with previous studies. In addition, we foundthat the baryon pole diagram, which is absent in the HB ChPT, can play an important role in certain cases.Compared with the results of other approaches, our predicted magnetic moments for the anti-triplets arerelatively small. The same is true for the Σ ++ c , Σ + c , and Ξ (cid:48) + c . On the other hand, our results for Σ c , Ξ (cid:48) c , and Ω c are relatively large (small in absolute value). It is not clear how to understand such a pattern at present.We hope that future lattice QCD or experimental studies can help us gain more insight into these importantquantities and better understand the singly charmed baryons. V. ACKNOWLEDGEMENTS
RXS thanks Jun-Xu Lu and Xiu-Lei Ren for useful discussions. This work is partly supported bythe National Natural Science Foundation of China under Grants No.11522539, No. 11735003, and thefundamental Research Funds for the Central Universities.
VI. APPENDIX
The pertinent loop functions, with the PCB terms removed, are given here. [1] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D , 030001 (2018). ( b ) B ¯3 ( δ , m φ ) = H ( b ) ( m ¯3 , δ , m φ ) , H ( d ) B ¯3 ( δ , m φ ) = H ( d ) ( m ¯3 , δ , m φ ) , H ( c ) B ¯3 ( δ , m φ ) = H ( c ) δ< m φ ( m ¯3 , δ , m φ ) , ( δ < m φ ) H ( c ) δ> m φ ( m ¯3 , δ , m φ ) , ( δ > m φ ) H ( e ) B ¯3 ( δ , m φ ) = H ( e ) δ< m φ ( m ¯3 , δ , m φ ) , ( δ < m φ ) H ( e ) δ> m φ ( m ¯3 , δ , m φ ) , ( δ > m φ ) H ( b ) B (0 , m φ ) = H ( b ) ( m , , m φ ) , H ( b ) B ( δ , m φ ) = H ( b ) ( m , − δ , m φ ) H ( d ) B (0 , m φ ) = H ( d ) ( m , , m φ ) , H ( d ) B ( δ , m φ ) = H ( d ) ( m , − δ , m φ ) H ( c ) B ( δ , m φ ) = H ( c ) δ< m φ ( m , δ , m φ ) , ( δ < m φ ) H ( e ) B ( δ , m φ ) = H ( e ) δ< m φ ( m , δ , m φ ) , ( δ < m φ ) (8) H ( b ) ( m B , , m φ ) = − π m φ + m φ m B (2 m B − m φ ) log m φ m B + m φ ( m φ − m φ m B + m B ) m B (cid:113) m B − m φ arccos( m φ m B ) , (9) H ( b ) ( m B , δ, m φ ) = m B π (cid:90) dx (cid:90) − x dy x m B + δ x m B x ( m B + δ ) + ( x − (cid:16) xm B − m φ (cid:17) + (cid:104)(cid:16) x + x − (cid:17) m B + δ (3 x − (cid:105) log x ( m B + δ ) + ( x − (cid:16) xm B − m φ (cid:17) µ − (cid:16) x + x − (cid:17) m B log x m B µ − x m B − δ + δ x , (0 < δ < m φ ) (10) H ( d ) ( m B , , m φ ) = − π m φ + m φ m B ( m B − m φ ) log m φ m B + m φ ( m φ − m B ) m B (cid:113) m B − m φ arccos( m φ m B ) , (11) H ( d ) ( m B , δ, m φ ) = − (2 m B + δ ) π m B (cid:40) (cid:104) m B (cid:16) m φ − δ (cid:17) + δ m B (cid:16) m φ − δ (cid:17) (cid:16) δ + m φ (cid:17) − (cid:16) m φ − δ (cid:17) (cid:105) log (cid:32) m φ m B + δ (cid:33) − (cid:118)(cid:117)(cid:117)(cid:116) (cid:16) m φ − δ (cid:17) (cid:16) δ + m φ (cid:17)(cid:16) m B + δ − m φ (cid:17) (cid:16) m B + δ + m φ (cid:17) (cid:104) m B (cid:16) δ − m φ (cid:17) + δ m B (cid:16) δ − m φ (cid:17) + δ m B + (cid:16) m φ − δ (cid:17) (cid:105) tan − − δ m B − m B − δ + m φ (cid:113)(cid:16) m φ − δ (cid:17) (cid:16) δ + m φ (cid:17) (cid:16) m B + δ − m φ (cid:17) (cid:16) m B + δ + m φ (cid:17) + (cid:104) m B (cid:16) δ − m φ (cid:17) + δ m B (cid:16) δ − m φ (cid:17) + δ m B + (cid:16) m φ − δ (cid:17) (cid:105) · (cid:118)(cid:117)(cid:117)(cid:116) (cid:16) m φ − δ (cid:17) (cid:16) δ + m φ (cid:17)(cid:16) m B + δ − m φ (cid:17) (cid:16) m B + δ + m φ (cid:17) · tan − m φ − δ (2 m B + δ ) (cid:113)(cid:16) m φ − δ (cid:17) (cid:16) δ + m φ (cid:17) (cid:16) m B + δ − m φ (cid:17) (cid:16) m B + δ + m φ (cid:17) + m B (cid:104) − δ m B + m B + (cid:16) m φ − δ (cid:17) (cid:16) δ + m φ (cid:17)(cid:105)(cid:111) + m B π , (12) ( c ) δ< m φ ( m B , δ, m φ ) = π m B ( δ + m B ) (cid:40) −
30 log (cid:32) δ + m B m φ (cid:33) m φ − (cid:34)(cid:32)
38 log (cid:32) m φ δ + m B (cid:33) + (cid:33) m B + tan − δ + m B δ − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) − tan − δ + m B δ + m B − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) · (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) − δ log (cid:32) δ + m B m φ (cid:33) (5 δ + m B ) (cid:35) m φ + − (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B (cid:17) δ + tan − δ + m B δ − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) − tan − δ + m B δ + m B − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) · (cid:16) δ + m B δ + m B (cid:17) + m B (cid:32) δ + m B δ + m B + (cid:32) m φ δ + m B (cid:33) · (cid:16) δ + m B δ + m B (cid:17)(cid:17)(cid:105) m φ + (cid:104)
30 log ( δ + m B ) (12 δ + m B ) m B − (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) m B + tan − δ + m B δ + m B − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) − tan − δ + m B δ − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) + (cid:32) − (cid:32) µ (cid:33) (3 δ + m B ) m B − (cid:16) m φ (cid:17) (22 δ + m B ) m B − δ log (cid:32) m φ δ + m B (cid:33) (38 δ + m B ) m B + δ log (cid:32) δ + m B m φ (cid:33) · (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17)(cid:17)(cid:105) m φ (cid:104) − δ log (cid:16) m φ (cid:17) m B ( δ + m B ) +
60 log ( δ + m B ) m B (2 δ + m B ) ( δ + m B ) + (cid:32) µ (cid:33) m B ( δ + m B ) (4 δ + m B ) + δ log (cid:32) m φ δ + m B (cid:33) m B (59 δ + m B ) − δ log (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B δ + m B (cid:17) + tan − δ + m B δ − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) − tan − δ + m B δ + m B − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) · (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) ( δ + m B ) (cid:16) δ + m B δ + m B δ + m B δ − m B (cid:17) + m B (cid:16) δ + m B δ + m B δ + m B δ + m B δ − m B δ − m B (cid:17)(cid:105)(cid:111) − m B (cid:18)
30 log (cid:18) m B µ (cid:19) − (cid:19) π , (13) ( c ) δ> m φ ( m B , δ, m φ ) = π m B ( δ + m B ) (cid:40) −
30 log (cid:32) δ + m B m φ (cid:33) m φ − (cid:34)(cid:32)
38 log (cid:32) m φ δ + m B (cid:33) + (cid:33) m B + − m φ + δ ( δ + m B ) (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) − δ log (cid:32) δ + m B m φ (cid:33) (5 δ + m B ) (cid:35) m φ + − (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B (cid:17) δ + − m φ + δ ( δ + m B ) (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) · (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) δ + m B δ + m B (cid:17) + m B (cid:16) δ + m B δ + m B + (cid:32) m φ δ + m B (cid:33) (cid:16) δ + m B δ + m B (cid:17)(cid:33)(cid:35) m φ + (cid:104)
30 log ( δ + m B ) m B − (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) m B − − m φ + δ ( δ + m B ) (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) · (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) + (cid:32)
20 log (cid:32) δ + m B m φ (cid:33) δ +
132 log (cid:32) δ + m B m φ (cid:33) m B δ +
316 log (cid:32) δ + m B m φ (cid:33) m B δ +
295 log (cid:32) δ + m B m φ (cid:33) m B δ − log (cid:32) m φ δ + m B (cid:33) m B (38 δ + m B ) δ +
360 log (cid:32) δ + m B m φ (cid:33) m B δ − (cid:32) ( δ + m B ) µ (cid:33) m B δ +
132 log (cid:32) δ + m B m φ (cid:33) m B δ − (cid:32) µ (cid:33) m B −
18 log (cid:16) m φ (cid:17) m B (cid:33)(cid:35) m φ + δ log ( δ + m B ) m B (2 δ + m B ) + − m φ + δ ( δ + m B ) (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) ( δ + m B ) (cid:16) δ + m B δ + m B δ + m B δ − m B ) + m B (cid:16) δ + m B δ + m B δ + m B δ + m B δ − m B δ − m B (cid:17) + (cid:34)
10 log (cid:32) ( δ + m B ) µ (cid:33) m B − δ log (cid:16) m φ (cid:17) ( δ + m B ) m B + δ log (cid:32) µ (cid:33) (4 δ + m B ) m B + δ log (cid:32) m φ δ + m B (cid:33) (59 δ + m B ) m B − δ log (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B δ + m B (cid:17)(cid:35)(cid:41) − m B (cid:18)
30 log (cid:18) m B µ (cid:19) − (cid:19) π , (14) ( e ) δ< m φ ( m B , δ, m φ ) = (cid:18)
80 log (cid:18) m B µ (cid:19) − (cid:19) m B π + π ( δ + m B ) m B (cid:40)(cid:34) − (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B (cid:17)(cid:35) m φ + (cid:34) − (cid:32) (cid:32)
28 log (cid:32) m φ δ + m B (cid:33) + (cid:33) δ + m B δ + m B (cid:33) m B + tan − δ + m B δ + m B − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) − tan − δ + m B δ − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) · (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) (cid:16) δ + m B δ + m B (cid:17) + (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17)(cid:35) m φ + (cid:34) δ log (cid:32) m φ δ + m B (cid:33) (cid:16) δ + m B δ + m B (cid:17) m B + (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) m B + tan − δ + m B δ − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) − tan − δ + m B δ + m B − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) − (cid:32) − δ log (cid:16) m φ (cid:17) m B +
60 log ( δ + m B ) (4 δ + m B ) m B + (cid:32) µ (cid:33) (3 δ + m B ) m B + δ log (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B δ + m B (cid:17)(cid:33)(cid:35) m φ − (cid:104)
20 log ( δ + m B ) (cid:16) − δ − m B δ + m B δ + m B (cid:17) m B + (cid:16) δ + m B δ + m B δ + m B δ + m B δ + m B δ − m B (cid:17) m B + tan − δ + m B δ − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) − tan − δ + m B δ + m B − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) · (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) ( δ + m B ) (cid:16) δ + m B δ + m B δ + m B δ + m B δ + m B (cid:17) + (cid:32) log (cid:32) m φ δ + m B (cid:33) m B (63 δ + m B ) δ − log (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) δ + log (cid:32) µ (cid:33) m B (cid:16) δ + m B δ + m B δ + m B (cid:17) + log (cid:16) m φ (cid:17) m B (cid:16) δ + m B δ + m B δ + m B (cid:17)(cid:33)(cid:35) m φ + (cid:34) δ log (cid:32) m φ δ + m B (cid:33) m B + δ log (cid:16) m φ (cid:17) m B (cid:16) δ + m B δ + m B δ + m B (cid:17) − δ log (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B δ + m B (cid:17) + δ tan − δ + m B δ − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) − tan − δ + m B δ + m B − m φ (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) (cid:113)(cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) m φ − δ (cid:17) ( δ + m B ) (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) − (cid:32) µ (cid:33) m B ( δ + m B ) (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) −
20 log ( δ + m B ) m B (cid:16) δ + m B δ + m B δ + m B δ + m B δ + m B (cid:17) + m B (cid:16) δ + m B δ + m B δ + m B δ + m B δ + m B δ + m B δ + m B δ + m B (cid:17)(cid:105)(cid:111) , (15) ( e ) δ> m φ ( m B , δ, m φ ) = (cid:18)
80 log (cid:18) m B µ (cid:19) − (cid:19) m B π + π ( δ + m B ) m B (cid:40)(cid:34) − (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B (cid:17)(cid:35) m φ + − (cid:32) m φ δ + m B (cid:33) (9 δ + m B ) δ − coth − δ + m B δ + m φ (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) (cid:16) δ + m B δ + m B (cid:17) · (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) + m B (cid:32) − δ − m B δ − m B + (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B (cid:17)(cid:33)(cid:35) m φ + (cid:34)(cid:32) δ + m B δ + m B δ +
16 log (cid:32) m φ δ + m B (cid:33) (cid:16) δ + m B δ + m B (cid:17) δ + m B δ + m B −
24 log m φ µ m B (3 δ + m B ) m B + − δ + m B δ + m φ (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) · (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) − (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17)(cid:35) m φ − (cid:104)(cid:16) δ + m B δ + m B δ + m B δ + m B δ + m B δ − m B (cid:17) m B + coth − δ + m B δ + m φ (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) ( δ + m B ) · (cid:16) δ + m B δ + m B δ + m B δ + m B δ + m B (cid:17) + (cid:32) µ (cid:33) (cid:16) m B + δ m B + δ m B (cid:17) +
20 log ( δ + m B ) (cid:16) m B + δ m B − δ m B (cid:17) + (cid:32) − log (cid:32) δ + m B m φ (cid:33) m B (cid:16) δ + m B δ + m B (cid:17) δ +
36 log m φ µ m B + log (cid:32) m φ δ + m B (cid:33) (cid:16) δ + m B δ + m B δ + m B (cid:17) δ + log (cid:16) m φ (cid:17) (cid:16) m B + δ m B + δ m B (cid:17)(cid:17)(cid:105) m φ + (cid:34) − (cid:32) log (cid:32) µ (cid:33) + δ + m B ) (cid:33) (23 δ + m B ) m B − δ log m φ µ (cid:16) δ + m B δ + m B δ + m B (cid:17) m B + δ log (cid:32) m φ δ + m B (cid:33) (cid:16) δ + m B (cid:17) m B + (cid:16) δ + m B δ + m B δ + m B δ + m B δ + m B δ + m B δ + m B δ + m B (cid:17) m B + δ coth − δ + m B δ + m φ (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) (cid:113)(cid:16) δ − m φ (cid:17) (cid:16) ( δ + m B ) − m φ (cid:17) · ( δ + m B ) (cid:16) δ + m B δ + m B δ + m B δ + m B (cid:17) − δ log (cid:32) δ + m B m φ (cid:33) (cid:16) δ + m B δ + m B δ + m B δ + m B δ + m B δ + m B (cid:17)(cid:35)(cid:41) . (16)
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