Abstract
The Bethe-Salpeter amplitude $\Phi(k,p)$ is expressed, by means of the Nakanishi integral representation, via a smooth function $g(\gamma,z)$. This function satisfies a canonical equation $g=Ng$. However, calculations of the kernel $N$ in this equation, presented previously, were restricted to one-boson exchange and, depending on method, dealt with complex multivalued functions. Although these difficulties are surmountable, but in practice, they complicate finding the unambiguous result. In the present work, an unambiguous expression for the kernel $N$ in terms of real functions is derived. For the one-boson scalar exchange, the explicit formula for $N$ is found. With this equation and kernel, the binding energies, calculated previously, are reproduced. Their finding, as well as calculation of the Bethe-Salpeter amplitude in the Minkowski space, become not more difficult than in the Euclidean one. The method can be generalized to any kernel given by irreducible Feynman graph. This generalization is illustrated by example of the cross-ladder kernel.
Highlights
One of the efficient methods of solving the BetheSalpeter (BS) equation [1] Φðk; pÞ 1⁄4i2 ×1⁄2ðp2Zþðdk2Þ4π2kÞ0−4 iKmð2kþ; ki0ε;p1⁄2ðÞp2Φ−ðkk0;Þ2p−Þ; m2 þ iεð1Þ directly in Minkowsky space, is based on using the Nakanishi representation [2] for the BS amplitude: ZZ Φðk; pÞ 1⁄4 −i 1 dz0 ∞ dγ0 −1 gðγ0; z0Þ × 1⁄2γ0 þ m2 − M2
We do not stop at the construction of the kernel Nðγ; z; γ0; z0Þ, but we reduce below Eq (5) to Eq (49) defined in the half-interval 0 ≤ z ≤ 1
We have found the new form of the kernel Nðγ; z; γ0; z0Þ of the canonical equation (5): g 1⁄4 Ng, for the Nakanishi function g for the ladder kernel
Summary
Ð1Þ directly in Minkowsky space, is based on using the Nakanishi representation [2] for the BS amplitude: ZZ Φðk; pÞ 1⁄4 −i 1 dz0 ∞ dγ0. Though the kernel N, derived below, might look a little bit lengthy and cumbersome, it is given by the explicit formulas, does not contain any integrations (for one-boson exchange), and it is easy, unambiguously and rapidly computated The advantages of this new representation of the kernel N in Eq (5) make solving the BS equation a simple and routine work. (9) and (10) and, the kernel Vðγ; z; γ0; z0Þ contains log and sqrt—the multivalued functions in the complex plane This requires, after substituting γ → γeiφ − z2m2 − ð1 − z2Þκ2, very careful definition of their branches. (7), (8), do not integrate over v in (8), but make the substitution γ → γeiφ − z2m2 − ð1 − z2Þκ2 in the argument γ In this way, the kernel (6) obtains the form. Calculating it, we will determine the integration limits v1;2
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