Erratum: Numerical values of the $f^F, f^D$ and $f^S$ coupling constants in SU(3) invariant Lagrangian of the interaction of the vector-meson nonets with $1/2^+$ octet baryons [Phys. Rev. C93, 055208 (2016)]

TL;DR

This paper clarifies that the numerical values of certain coupling constants in an SU(3) invariant Lagrangian are unaffected by the specific $ ext{omega-phi}$ mixing scheme used.

Contribution

It provides a correction or clarification showing the independence of coupling constants from the $ ext{omega-phi}$ mixing choice in the SU(3) invariant framework.

Findings

Coupling constants are independent of $ ext{omega-phi}$ mixing scheme.

The paper corrects previous assumptions about mixing dependence.

Clarifies the theoretical consistency of the coupling constants.

Abstract

It is clearly demonstrated that numerical values of the $f^F, f^D$ and $f^S$ coupling constants in SU(3) invariant Lagrangian of the interaction of the vector-meson nonets with $1/2^+$ octet baryons do not depend on the chosen version of the $\omega-\phi$ mixing.

Equations

$$L_{VB\bar{B}}=\frac{i}{\sqrt{2}}{f}^F[{\bar{B}}_{\beta }^{\alpha }{\gamma }_{\mu }{B}_{\gamma }^{\beta }-{\bar{B}}_{\gamma }^{\beta }{\gamma }_{\mu }{B}_{\beta }^{\alpha }](V^{\mu }{)}_{\alpha }^{\gamma }+$$

$$+\frac{i}{\sqrt{2}}{f}^D[{\bar{B}}_{\beta }^{\alpha }{\gamma }_{\mu }{B}_{\gamma }^{\beta }+{\bar{B}}_{\gamma }^{\beta }{\gamma }_{\mu }{B}_{\beta }^{\alpha }](V^{\mu }{)}_{\alpha }^{\gamma }+$$

$$+\frac{i}{\sqrt{2}}{f}^S{\bar{B}}_{\beta }^{\alpha }{\gamma }_{\mu }{B}_{\alpha }^{\beta }{\omega }_0^{\mu }$$

$$1.\varphi$$

$$\omega$$

$$2.\varphi$$

$$\omega$$

$$3.\varphi$$

$$\omega$$

$$4.\varphi$$

$$\omega$$

$$1.f^F$$

$$f^D$$

$$f^S$$

$$2.f^F$$

$$f^D$$

$$f^S$$

$$3.f^F$$

$$f^D$$

$$f^S$$

$$4.f^F$$

$$f^D$$

$$f^S$$

$$1.\frac{1}{f_{\rho }}:\frac{1}{f_{\omega }}:\frac{1}{f_{\varphi }}$$

$$2.\frac{1}{f_{\rho }}:\frac{1}{f_{\omega }}:\frac{1}{f_{\varphi }}$$

$$3.\frac{1}{f_{\rho }}:\frac{1}{f_{\omega }}:\frac{1}{f_{\varphi }}$$

$$4.\frac{1}{f_{\rho }}:\frac{1}{f_{\omega }}:\frac{1}{f_{\varphi }}$$

$${\omega }_8$$

$${\omega }_0$$

$$\sin \theta =\sqrt{\frac{1}{3}};\cos \theta =\sqrt{\frac{2}{3}},$$

$$\varphi =-\bar{s}s$$

$$\omega =+\frac{1}{\sqrt{2}}(\bar{u}u+\bar{d}d).$$

$$J_{\mu }^h=\frac{2}{3}\bar{u}{\gamma }_{\mu }u-\frac{1}{3}\bar{d}{\gamma }_{\mu }d-\frac{1}{3}\bar{s}{\gamma }_{\mu }s,$$

$$J_{\mu }^h=\frac{1}{2}(\bar{u}{\gamma }_{\mu }u-\bar{d}{\gamma }_{\mu }d)+\frac{1}{6}(\bar{u}{\gamma }_{\mu }u+\bar{d}{\gamma }_{\mu }d)-\frac{1}{3}(\bar{s}{\gamma }_{\mu }s),$$

$$J_{\mu }^h=\frac{1}{\sqrt{2}}{J}_{\mu }^{{\rho }^0}+\frac{1}{3\sqrt{2}}{J}_{\mu }^{\omega }+\frac{1}{3}{J}_{\mu }^{\varphi }.$$

$$J_{\mu }^h=\frac{m_{{\rho }^0}^2}{f_{\rho }}{\rho }_{\mu }^0+\frac{m_{\omega }^2}{f_{\omega }}{\omega }_{\mu }+\frac{m_{\varphi }^2}{f_{\varphi }}{\varphi }_{\mu },$$

$$\frac{1}{f_{\rho }}=A\frac{1}{\sqrt{2}};\frac{1}{f_{\omega }}=+A\frac{1}{3\sqrt{2}};\frac{1}{f_{\varphi }}=+A\frac{1}{3},$$

$$\frac{1}{f_{\rho }}:\frac{1}{f_{\omega }}:\frac{1}{f_{\varphi }}$$

$$\varphi =+\bar{s}s$$

$$\omega =+\frac{1}{\sqrt{2}}(\bar{u}u+\bar{d}d).$$

$$J_{\mu }^h=\frac{1}{\sqrt{2}}{J}_{\mu }^{{\rho }^0}+\frac{1}{3\sqrt{2}}{J}_{\mu }^{\omega }-\frac{1}{3}{J}_{\mu }^{\varphi },$$

Full text

Erratum: Numerical values of the $f^{F},f^{D}$ and $f^{S}$ coupling constants in SU(3)

invariant Lagrangian of the interaction of the vector-meson nonets with $1/2^{+}$ octet baryons [Phys. Rev. C93, 055208 (2016)]

Cyril Adamuščin

Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovak Republic

  

Erik Bartoš

Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovak Republic

  

Stanislav Dubnička

Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovak Republic

  

Anna Z. Dubničková

Department of Theoretical Physics, Comenius University, Bratislava, Slovak Republic

Some aspects of the presented determination of numerical values of the $f^{F},f^{D}$ and $f^{S}$ coupling constants in the SU(3) invariant interaction Lagrangian

[TABLE]

of the vector-meson nonet with $1/2^{+}$ octet baryons in abdd were not correct.

The crucial ingredient in that determination is an application of the $\omega-\phi$ mixing, which has been entered into the procedure in abdd two times. First in a derivation of expressions (51), (52), (55) and (56) for coupling constants of vector-mesons with nucleons from the above-mentioned Lagrangian, and so, also in a derivation of the reverse expressions (66), (67), (68) and (69), and then in a determination of the signs of the universal vector-meson coupling constants $f_{\rho},f_{\omega}$ and $f_{\phi}$.

Generally there are in literature the following four different physically acceptable forms of the $\omega-\phi$ mixing

[TABLE]

which manifest themselves in the four different forms of expressions for $f^{F},f^{D},f^{S}$

[TABLE]

On the other hand an application of the same $\omega-\phi$ mixing configurations (1), (2), (3), (4) leads to the rates of the overthrown values of the universal vector-meson coupling constants with different signs as follows

[TABLE]

which can be found e.g. in ref. perren on p. 52 relation (A)

[TABLE]

to be found in ref. clcott on p. 539

[TABLE]

and this last case can be found e.g. in ref. gas on p. 446.

All these relations can be explained by the following considerations.

Starting e.g. from the $\omega-\phi$ mixing configuration (1) and substituting explicitly

[TABLE]

and for the ideal mixing angle $\theta=35.3^{0}$

[TABLE]

one obtains

[TABLE]

and

[TABLE]

On the other hand the hadronic electromagnetic (EM) current

[TABLE]

can be formally arranged to the shape

[TABLE]

and because

$\frac{1}{\sqrt{2}}(\bar{u}\gamma_{\mu}u-\bar{d}\gamma_{\mu}d)=J^{\rho^{0}}_{\mu}$

$+\frac{1}{\sqrt{2}}(\bar{u}\gamma_{\mu}u+\bar{d}\gamma_{\mu}d)=J^{\omega}_{\mu}$ due to the sign ”+” in (16)

$-\bar{s}\gamma_{\mu}s=J^{\phi}_{\mu}$ due to the sign ”-” in (15),

are the $\rho^{0}-,\omega-,\phi-$ meson EM currents, respectively, the hadronic EM current acquires the following form

[TABLE]

Now, if the results of the Kroll-Lee-Zumino paper klz , that a linear combination of the neutral vector-meson fields $\rho^{0}_{\mu},\omega_{\mu},\phi_{\mu}$

[TABLE]

with the universal vector-meson coupling constants $f_{\rho},f_{\omega},f_{\phi}$, is proportional by some real constant $A$ to the hadronic EM current (18), are taken into account, considering a dimension of the Dirac quark fields $u,d,s$ in (18), in the framework of the natural units $\hbar=c=1$, to be m${}^{\frac{3}{2}}$ and a dimension of the vector-meson fields in (19) to be m1, the relations

[TABLE]

are found.

Then from these last relations the rates

[TABLE]

are obtained, giving the signs of the universal vector-meson coupling constants $+f_{\rho},+f_{\omega},+f_{\phi}$.

As a matter of fact the signs of the universal vector-meson coupling constants are specified already from the relations (20), however, a community of physicists prefers the relations (9)-(12), in which the universal vector-meson coupling constants are related to $\sin\theta$ and $\cos\theta$ where the angle $\theta$ is determined from the quadratic Gell-Mann-Okubo vector-meson mass formula, which provide more realistic values of the latter to be in fair agreement with experimental evaluations. Therefore we also favor a presentation of the universal vector-meson coupling constants signs in the form (9)-(12).

In a like manner, starting from the $\omega-\phi$ mixing configuration (2), and relations (13) and (14), one obtains

[TABLE]

and

[TABLE]

Then comparing the hadronic EM current to be multiplied by the real constant $A$

[TABLE]

where

$J^{\rho^{0}}_{\mu}=\frac{1}{\sqrt{2}}(\bar{u}\gamma_{\mu}u-\bar{d}\gamma_{\mu}d)$

$J^{\omega}_{\mu}=+\frac{1}{\sqrt{2}}(\bar{u}\gamma_{\mu}u+\bar{d}\gamma_{\mu}d)$ due to the sign ”+” in (22)

$J^{\phi}_{\mu}=+\bar{s}\gamma_{\mu}s$ due to the sign ”+” in (21),

with (19), one obtains relations

[TABLE]

and from them the rates (10), giving the following signs of universal vector-meson coupling constants $+f_{\rho},+f_{\omega},-f_{\phi}$.

Again, starting from the $\omega-\phi$ mixing configuration (3), and relations (13) and (14), one obtains

[TABLE]

and

[TABLE]

Then comparing the hadronic EM current to be multiplied by the real constant $A$

[TABLE]

where

$J^{\rho^{0}}_{\mu}=\frac{1}{\sqrt{2}}(\bar{u}\gamma_{\mu}u-\bar{d}\gamma_{\mu}d)$

$J^{\omega}_{\mu}=-\frac{1}{\sqrt{2}}(\bar{u}\gamma_{\mu}u+\bar{d}\gamma_{\mu}d)$ due to the sign ”-” in (26)

$J^{\phi}_{\mu}=+\bar{s}\gamma_{\mu}s$ due to the sign in (25),

with (19), one obtains relations

[TABLE]

and from them the rates (11), giving the following signs of universal vector-meson coupling constants $+f_{\rho},-f_{\omega},-f_{\phi}$.

Finally, starting from the $\omega-\phi$ mixing configuration (4), and relations (13) and (14), one obtains

[TABLE]

and

[TABLE]

Then comparing the hadronic EM current to be multiplied by the real constant $A$

[TABLE]

where

$J^{\rho^{0}}_{\mu}=\frac{1}{\sqrt{2}}(\bar{u}\gamma_{\mu}u-\bar{d}\gamma_{\mu}d)$

$J^{\omega}_{\mu}=-\frac{1}{\sqrt{2}}(\bar{u}\gamma_{\mu}u+\bar{d}\gamma_{\mu}d)$ due to the sign ”-” in (30)

$J^{\phi}_{\mu}=-\bar{s}\gamma_{\mu}s$ due to the sign ”-” in (29),

with (19), one obtains relations

[TABLE]

and from them the rates (12), giving the following signs of universal vector-meson coupling constants $+f_{\rho},-f_{\omega},+f_{\phi}$.

The signs of the universal vector-meson coupling constants $f_{\rho},f_{\omega},f_{\phi}$ are very important to be known, as the numerical values of these constants are regularly estimated from the experimental values olive of the vector-meson lepton widths by means of the formula

[TABLE]

in which $f_{V}$ is contained in a quadratic form.

From (5), (6), (7), (8) it seems at first sight that numerical values of the coupling constants $f^{F},f^{D},f^{S}$ have to depend on the choice of the $\omega-\phi$ mixing version.

However, if in (5), (6), (7), (8) the coupling constants of vector-mesons with nucleons are determined e.g from $(f^{(1)}_{\rho NN}/f_{\rho}),(f^{(1)}_{\omega NN}/f_{\omega}),(f^{(1)}_{\phi NN}/f_{\phi})$, to be found in a fitting procedure of all existing data on nucleon EM structure, by means of the signs of $f_{\rho},f_{\omega},f_{\phi}$ following from (9), (10), (11), (12), as it is demonstrated above, then from all four different expressions in (5), (6), (7), (8) one obtains the same numerical values for $f^{F}_{1},f^{D}_{1},f^{S}_{1}$ as follows

[TABLE]

By means of a similar procedure one can find also numerical values of all other coupling constants under consideration

[TABLE]

In the paper abdd for the numerical evaluation of the vector-meson-nucleon coupling constants from the $(f_{\rho NN}/f_{\rho}),(f_{\omega NN}/f_{\omega}),(f_{\phi NN}/f_{\phi})$ we have applied the signs of the universal vector-meson coupling constants (10) (to be inspired by Close and Cottingham in clcott ), which moreover have been combined with expressions for $f^{F},f^{D},f^{S}$

[TABLE]

to be generated by the physically non-acceptable form of the $\omega-\phi$ mixing

[TABLE]

used by Gasiorovicz gas on p.325, leading to the numerical values of $f^{F},f^{D},f^{S}$ differing from the correct values presented in this Erratum.