Mass Spectrum of Pseudo-Scalar Glueballs from a Bethe–Salpeter Approach with the Rainbow–Ladder Truncation

Abstract

We suggest a framework based on the rainbow approximation to the Dyson–Schwinger and Bethe–Salpeter equations with effective parameters adjusted to lattice QCD data to calculate the masses of the ground and excited states of pseudo-scalar glueballs. The structure of the truncated Bethe–Salpeter equation with the gluon and ghost propagators as solutions of the truncated Dyson–Schwinger equations is analyzed in Landau gauge. Both, the Bethe–Salpeter and Dyson–Schwinger equations, are solved numerically within the same rainbow–ladder truncation with the same effective parameters which ensure consistency of the approach. We found that with a set of parameters, which provides a good description of the lattice data within the Dyson–Schwinger approach, the solutions of the Bethe–Salpeter equation for the pseudo-scalar glueballs exhibit a rich mass spectrum which also includes the ground and excited states predicted by lattice calculations. The obtained mass spectrum contains also several intermediate excitations beyond the lattice approaches. The partial Bethe–Salpeter amplitudes of the pseudo-scalar glueballs are presented as well.

Appendices

Appendix A: Partial Decomposition of the Rainbow Kernel

The spatial dependence of the integrand on $\varOmega _{{\mathbf {k}}}$ is contained in the rainbow exponents and in the scalar product $(p\cdot k)^\delta = ({\tilde{p}}\ {\tilde{k}} \ x_{kp})^\delta$,

$$\begin{aligned} \exp (\alpha x_{kp}) x_{kp}^\delta = \sum _{M_v} W_{M_v}^{(\delta )}({\tilde{p}},{\tilde{k}}) G_{M_v}^{(1)}(x_{kp}), \end{aligned}$$

(26)

where $\alpha =2{\tilde{k}}{\tilde{p}}/\omega ^2$. The partial coefficients $W_{M_v}^{(\delta )}({\tilde{p}},{\tilde{k}})$ can be computed explicitly as

$$\begin{aligned} W_{M_v}^{(\delta )}({\tilde{p}},{\tilde{k}})= & {} \frac{2}{\pi }\int _{-1}^1 \sqrt{1-x_{kp}^2} \exp (\alpha x_{kp}) x_{kp}^\delta G_{M_v}^{(1)}(x_{kp})d x_{kp} \nonumber \\= & {} \frac{2}{\pi }\frac{d^\delta }{d\alpha ^\delta }\left[ \int _{-1}^1 \sqrt{1-x_{kp}^2} \mathrm{e}^{\alpha x_{kp}} G_{M_v}^{(1)}(x_{kp})\right] d x_{kp} = 2(M_v+1) \frac{d^\delta }{d\alpha ^\delta } \left[ \frac{1}{\alpha }I_{M_v+1}(\alpha )\right] , \end{aligned}$$

(27)

where $I_{M_v+1}(\alpha )$ are the modified Bessel functions of second kind (of the imaginary argument) yielding

$$\begin{aligned} \mathrm{e}^{\alpha x_{kp}} x_{kp}^\delta =2 \sum _{M_v} (M_v+1) \frac{d^\delta }{d\alpha ^\delta }\left[ \frac{1}{\alpha }I_{M_v+1}(\alpha )\right] G_{M_v}^{(1)}(x_{kp}). \end{aligned}$$

(28)

The dependence on the spatial angles of the vectors p and k enters via $G_{M_v}^{(1)}(x_{kp}\equiv \cos \xi _{kp})$, where $\cos \xi _{kp}=\cos \xi _p\cos \xi _k + \sin \xi _p\sin \xi _k\cos \theta _{{\mathbf {kp}}}$. Explicitly, such a dependence can be written by using an addition theorem for Gegenbauer polynomials

$$\begin{aligned} G_{M_v}^{(1)}(x)=\frac{2\pi ^2}{M_v+1} \sum _{l\mu } Z_{M_v l\mu }^*(p) Z_{M_v l\mu }(k) \end{aligned}$$

(29)

with $Z_{M_v l\mu }(k)=Z_{M_v l\mu }(\xi _k,\theta _{{\mathbf {k}}},\phi _{{\mathbf {k}}})$ as hyper-spherical harmonics, to obtain

$$\begin{aligned} \mathrm{e}^{\alpha x_{kp}} x_{kp}^\delta =4\pi ^2 \sum _{M_v,l,\mu } \frac{d^\delta }{d\alpha ^\delta } \left[ \frac{1}{\alpha }I_{M_v+1}(\alpha )\right] Z_{M_v l\mu }^*(p) Z_{M_v l\mu }(k), \end{aligned}$$

(30)

where the normalized hyperspherical harmonics are $Z_{M_v l\mu }(p) = X_{M_v l}(\xi _p) \mathrm{Y}_{l\mu }({\mathbf {p}})$ with

$$\begin{aligned} X_{M_v l}(\xi _p) = 2^l l!\sqrt{\frac{2}{\pi }} \sqrt{\frac{(M_v+1)(M_v-l)!}{(M_v+l+1)!}} \sin ^l\xi _p G_{M_v-l}^{l+1}(\cos \xi _p). \end{aligned}$$

(31)

At a first glance, Eqs. (26)–(31) seemingly even complicate the integration. However, by observing that the dependence of the integrand in (6) on the spatial angles $\varOmega _{{\mathbf {k}}}$ is only through the interaction kernel and trough $x_{kp}^\delta$, Eq. (30), i.e. only trough the spatial harmonics $\mathrm{Y}_{l\mu }({\mathbf {k}})$, the integration over $d\varOmega _{{\mathbf {k}}}$ is trivial and eventually we have

$$\begin{aligned} \int \mathrm{e}^{\alpha x_{kp}} x_{kp}^\delta d\varOmega _{{\mathbf {k}}} = 8\pi \sum _{M_v}\frac{d^\delta }{d\alpha ^\delta } \left[ \frac{1}{\alpha }I_{M_v+1}(\alpha )\right] G_{M_v}^{(1)}(x_k) G_{M_v}^{(1)}(x_p). \end{aligned}$$

(32)

Appendix B: Integration over $x_k$

Here, we present some details of the integration over the hyper angle $x_k$ and the resulting explicit expressions of selection rules. The corresponding angular integral is of the form

$$\begin{aligned} {{\mathcal {K}}}_{M_v,M_k}^{L}= \int \limits _{-1}^1\sqrt{1-x_k^2} x_k^L G_{M_k}^{(1)}(x)G_{M_v}^{(1)}(x_k) dx_k. \end{aligned}$$

(33)

Due to parity restrictions, the partial amplitudes $F_{M_k}$ contain only even values of the Gegenbauer polynomials, i.e. $M_k=[0,2,4,.. M^{max}]$, where $M^{max}$ is the maximum number of polynomials taken into account in concrete calculations. The Gegenbauers $G_{M_v}^{(1)}(x_k)$, which come from the interaction kernel (32), may contain both, even and odd values of $M_v$, and formally the summation is extended to infinite, $M_v=[0..\infty ]$. However, not all values in this interval contribute to (33). The symmetrical limits of integration restrict the Gegenbauer polynomials in (33) to obey the condition ($L+M_k+M_v$)=even. Other restrictions originate from the explicit expression of the integral, see below. From a standard math handbook one infers that

$$\begin{aligned}&\int \limits _{-1}^1\sqrt{1-x^2} x^L G_{M_k}^{(\lambda )}(x)G_{M_v}^{(\lambda )}(x) dx = 2^{M_k+M_v}\frac{(2\lambda )_{M_k} (2\lambda )_{M_v} L!}{m_k!M_v!(L+M_k+M_v)!}\left( \frac{L-M_k-M_v}{2}+1\right) _{M_k+M_v} \nonumber \\&\times B\left( \lambda +\frac{1}{2},\frac{L+M_k+M_v+1}{2}\right) {}_3F_2\left( -M_k,-M_v,1;2\lambda ,\frac{L-M_k-M_v}{2}+1;1\right) , \end{aligned}$$

(34)

where ${}_3F_2$ is the generalized hypergeometric function and $(a)_k=a (a+1)(a+2)\ldots (a+k-1)$ (with $a_0=1$) is the known Pochhammer symbol and $B(x,y)=\frac{\varGamma (x)\varGamma (y)}{\varGamma (x+1)}$ and $\varGamma (x)$ are the familiar Euler $\beta -$ and $\varGamma -$ functions, respectively. Despite the integral (34) is finite, at some values of $L,M_k$ and $M_v$ the product of the Pochhammer symbol and hypergeometric function can be of the type $0 \cdot \infty$, which implies that Eq. (34) cannot be implemented directly into numerical calculations. One needs to handle zeros and singularities manually. We use the obvious properties

$$\begin{aligned} (-m)_k=(-1)^k\frac{ m!}{(m-k)!}; \qquad (a)_k=\frac{\varGamma (a+k)}{\varGamma (a)} \end{aligned}$$

(35)

to obtain

$$\begin{aligned}&\left( \frac{L-M_k-M_v}{2}+1\right) _{M_k+M_v}=\frac{\kappa !}{\kappa _1!},\end{aligned}$$

(36)

$$\begin{aligned}&B\left( \frac{3}{2},\frac{L+M_k+M_v+1}{2}\right) =\frac{\pi }{2^{\kappa +1}}\frac{(2\kappa )!!}{(\kappa +1)!}, \end{aligned}$$

(37)

$$\begin{aligned} {}_3F_2\left( -M_k,-M_v,1;2,\frac{L-M_k-M_v}{2}+1;1\right) = \sum _{k=0}^\infty \frac{1}{(k+1)!} \frac{M_k!M_v!}{(M_k-k)!(M_v-k)!} \frac{\kappa _1!}{(\kappa _1+k)!}, \end{aligned}$$

where, for brevity, we introduce the shorthand notation $\kappa =\frac{L+M_k+M_v}{2}$, $\kappa _1=\frac{L-M_k-M_v}{2}$.

With this notation, the integral (34) reads

$$\begin{aligned} \int \limits _{-1}^1\sqrt{1-x^2} x^L G_{M_k}^{(1)}(x)G_{M_v}^{(1)}(x) dx= & {} \pi \frac{2^{M_k+M_v-\kappa -1} (M_k+1)!(M_v+1)!L!}{(\kappa +1)(2\kappa )!!} \nonumber \\\times & {} \sum _{k=0}^\infty \frac{1}{(k+1)!(M_k-k)!(M_v-k)!(\kappa _1+k)!}. \end{aligned}$$

(38)

In Eq. (38) the summation is restricted by those values of k which ensure non-negative factorials, i.e. in the above sum $k\le M_k$ and $k\le M_v$ and $(M_k+M_v-L+2k)\ge 0$. Together with the condition $(L+M_k+M_v)$-even, these restrictions form the selecting rules for the integral (33). Actually, in practice the summation in (38) consists only of one, or, at maximum, two terms. Consequently, the integrals (33) turn out to be extremely simple, being expressed in form of the fractional parts of $\pi$. For instance, the value L=0 results in the orthogonal condition for the Gegenbauer polynomials i.e. ${{\mathcal {K}}}_{M_k,M_v}^{L=0}=\frac{\pi }{2}\delta _{M_k,M_v}$. For $L=1$ one has $M_v=1,3,5,7$ and for even $M_k=0,2,4,6,8$ the integral (33) is always $\pi /4$. Here we present the corresponding explicit expressions of the integrals (33) for $M_k^{max}=8$:

$$\begin{aligned}&{{\mathbf {L=0}}}:\qquad {{\mathcal {K}}}_{M_k,M_v}^{L=0}=\frac{\pi }{2} \delta _{M_k,M_v},\end{aligned}$$

(39)

$$\begin{aligned}&{{\mathbf {L=1}}}: \qquad M_k=0 \rightarrow {{\mathcal {K}}}_{0,1}^{L=1}=\frac{\pi }{4};\quad {{\mathcal {K}}}_{0,3}^{L=1}=0,\end{aligned}$$

(40)

$$\begin{aligned}&M_k=2 \rightarrow {{\mathcal {K}}}_{2,1}^{L=1}={{\mathcal {K}}}_{2,3}^{L=1}=\frac{\pi }{4},\end{aligned}$$

(41)

$$\begin{aligned}&M_k=4 \rightarrow {{\mathcal {K}}}_{4,3}^{L=1}={{\mathcal {K}}}_{4,5}^{L=1}=\frac{\pi }{4},\end{aligned}$$

(42)

$$\begin{aligned}&M_k=6 \rightarrow {{\mathcal {K}}}_{6,5}^{L=1}={{\mathcal {K}}}_{6,7}^{L=1}=\frac{\pi }{4},\end{aligned}$$

(43)

$$\begin{aligned}&M_k=8 \rightarrow {{\mathcal {K}}}_{8,7}^{L=1}={{\mathcal {K}}}_{8,9}^{L=1}=\frac{\pi }{4},\end{aligned}$$

(44)

$$\begin{aligned}&{{\mathbf {L=2}}}:\qquad M_k=0 \rightarrow {{\mathcal {K}}}_{0,0}^{L=2}={{\mathcal {K}}}_{0,2}^{L=2}=\frac{\pi }{8},\end{aligned}$$

(45)

$$\begin{aligned}&M_k=2 \rightarrow {{\mathcal {K}}}_{2,0}^{L=2}=\frac{\pi }{8}; \quad {{\mathcal {K}}}_{2,2}^{L=2}=\frac{\pi }{4};\quad {{\mathcal {K}}}_{2,4}^{L=2}=\frac{\pi }{8},\end{aligned}$$

(46)

$$\begin{aligned}&M_k=4 \rightarrow {{\mathcal {K}}}_{4,2}^{L=2}=\frac{\pi }{8}; \quad {{\mathcal {K}}}_{4,4}^{L=2}=\frac{\pi }{4};\quad {{\mathcal {K}}}_{4,6}^{L=2}=\frac{\pi }{8},\end{aligned}$$

(47)

$$\begin{aligned}&M_k=6 \rightarrow {{\mathcal {K}}}_{6,4}^{L=2}=\frac{\pi }{8}; \quad {{\mathcal {K}}}_{6,6}^{L=2}=\frac{\pi }{4};\quad {{\mathcal {K}}}_{6,8}^{L=2}=\frac{\pi }{8},\end{aligned}$$

(48)

$$\begin{aligned}&M_k=8 \rightarrow {{\mathcal {K}}}_{8,6}^{L=2}=\frac{\pi }{8}; \quad {{\mathcal {K}}}_{8,8}^{L=2}=\frac{\pi }{4};\quad {{\mathcal {K}}}_{8,10}^{L=2}=\frac{\pi }{8},\end{aligned}$$

(49)

$$\begin{aligned}&{{\mathbf {L=3}}}:\qquad M_k=0 \rightarrow {{\mathcal {K}}}_{0,1}^{L=3}=\frac{\pi }{8};\quad {{\mathcal {K}}}_{0,3}^{L=3}=\frac{\pi }{16},\end{aligned}$$

(50)

$$\begin{aligned}&M_k=2 \rightarrow {{\mathcal {K}}}_{2,1}^{L=3}=\frac{3\pi }{16};\quad {{\mathcal {K}}}_{2,3}^{L=3}=\frac{3\pi }{16};\quad {{\mathcal {K}}}_{2,5}^{L=3}=\frac{\pi }{16},\end{aligned}$$

(51)

$$\begin{aligned}&M_k=4 \rightarrow {{\mathcal {K}}}_{4,1}^{L=3}=\frac{\pi }{16};\quad {{\mathcal {K}}}_{4,3}^{L=3}=\frac{3\pi }{16};\quad {{\mathcal {K}}}_{4,5}^{L=3}=\frac{3\pi }{16}; \quad K_{4,7}^{L=3}=\frac{\pi }{16},\end{aligned}$$

(52)

$$\begin{aligned}&M_k=6 \rightarrow {{\mathcal {K}}}_{6,3}^{L=3}=\frac{\pi }{16};\quad {{\mathcal {K}}}_{6,5}^{L=3}=\frac{3\pi }{16};\quad {{\mathcal {K}}}_{6,7}^{L=3}=\frac{3\pi }{16}; \quad {{\mathcal {K}}}_{6,9}^{L=3}=\frac{\pi }{16},\end{aligned}$$

(53)

$$\begin{aligned}&M_k=8 \rightarrow {{\mathcal {K}}}_{8,5}^{L=3}=\frac{\pi }{16};\quad {{\mathcal {K}}}_{8,7}^{L=3}=\frac{3\pi }{16},\quad {{\mathcal {K}}}_{8,9}^{L=3}=\frac{3\pi }{16}; \quad {{\mathcal {K}}}_{8,11}^{L=3}=\frac{\pi }{16}. \end{aligned}$$

(54)

Appendix C: Explicit Expressions for the Coefficients $C_{L,\delta }({\tilde{k}},{\tilde{p}},x_p)$ in Eq. (22)

Recall that $M_p$ comes from the decomposition of the amplitude in l.h.s., $M_k$ is the decomposition of the BS amplitude under the integral in the r.h.s. of Eq. (22), and $M_v$ comes from decomposition of the rainbow interaction. Eventually, $\delta$ corresponds to the power of the scalar product $(k\cdot p)^\delta$ and L is the power of $k_4^L$ which we meet after contractions over the Lorentz indices. Results of the contraction provide $L=0,1,2,3$ and $\delta =0,1,2,3)$ and that $L+\delta _{max}=3$. It implies that not all possible combination of $L,\delta$ in $C_{L,\delta }({\tilde{k}},{\tilde{p}})$ contribute to the r.h.s. There are 10 possible combinations: $(L,\delta ) =(0,0)$, $(L,\delta ) =(0,1)$, $(L,\delta ) =(0,2)$, $(L,\delta ) =(0,3)$, $(L,\delta ) =(1,0)$, $(L,\delta ) =(1,1)$, $(L,\delta ) =(1,2)$, $(L,\delta ) =(2,0)$, $(L,\delta ) =(2,1)$, $(L,\delta ) =(3,0)$. Explicitly one has

$$\begin{aligned} C_{0,0}({\tilde{p}},{\tilde{k}},x_p)&= 96 {\tilde{k}}^2{\tilde{p}}^2 (1 -x_p^2) M_{gg}^2 ({\tilde{k}}^2 +{\tilde{p}}^2); \quad C_{0,1}({\tilde{p}},{\tilde{k}},x_p) -12 M_{gg}^2 {\tilde{k}} {\tilde{p}}\nonumber \\&\qquad \qquad \times (-M_{gg}^ 2 {\tilde{k}}^2 - M_{gg}^2 {\tilde{p}}^2 + M_{gg}^2 {\tilde{p}}^2 x_p^2 + 12{\tilde{k}}^2 {\tilde{p}}^2 - 16 {\tilde{k}}^2{\tilde{p}}^2 x_p^2),\nonumber \\ C_{0,2}({\tilde{p}},{\tilde{k}},x_p)&= -24 M_{gg}^2 {\tilde{k}}^2{\tilde{p}}^2 \ (M_{gg}^2 + 4 {\tilde{k}}^2 + 4{\tilde{p}}^2);\quad C_{0,3}({\tilde{p}},{\tilde{k}},x_p)= 144 M_{gg}^2{\tilde{k}}^3{\tilde{p}}^3, \end{aligned}$$

(55)

$$\begin{aligned} C_{1,0}({\tilde{p}},{\tilde{k}},x_p)&=12 M_{gg}^2{\tilde{k}}{\tilde{p}} x_p \ (-M_{gg}^2 {\tilde{k}}^2-M_{gg}^2 {\tilde{p}}^2+M_{gg}^2 {\tilde{p}}^2 x_p^2 -4{\tilde{k}}^2{\tilde{p}}^2), \nonumber \\&\qquad \qquad C_{1,1}({\tilde{p}},{\tilde{k}},x_p) =48 M_{gg}^2{\tilde{k}}^2{\tilde{p}}^2 x_p, \ (M_{gg}^2 + 4 {\tilde{k}}^2 + 4{\tilde{p}}^2), \nonumber \\ C_{1,2}({\tilde{p}},{\tilde{k}},x_p)&= -336{\tilde{p}}^3 x_p M_{gg}^2 {\tilde{k}}^3, \end{aligned}$$

(56)

$$\begin{aligned} C_{2,0}({\tilde{p}},{\tilde{k}},x_p)&=-24 M_{gg}^2{\tilde{k}}^2 \ (M_{gg}^2 {\tilde{p}}^2*x_p^2+4 {\tilde{k}}^2* {\tilde{p}}^2 + 4{\tilde{p}}^4),\quad C_{2,1}({\tilde{p}},{\tilde{k}},x_p) = -12 {\tilde{p}} M_{gg}^2 {\tilde{k}}^3 (M_{gg}^2 - 16 {\tilde{p}}^2), \end{aligned}$$

(57)

$$\begin{aligned} C_{3,0}({\tilde{p}},{\tilde{k}},x_p)&=12 M_{gg}^4 {\tilde{k}}^3 {\tilde{p}} \ x_p. \end{aligned}$$

(58)

Actually, expressions (55–58) imply that the summation $\sum _{L,\delta }$ in Eq. (22) is restricted to $\sum _{L=0}^3\sum _{\delta =0}^{\delta _{max}}$, where $L+\delta _{max}=3$.