Accepted Manuscript Pion wave-function as a bound system in 3+1 dimensional QCD Teruo Kurai PII: DOI: Reference:
S2211-3797(17)30081-5 http://dx.doi.org/10.1016/j.rinp.2017.05.028 RINP 713
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
15 January 2017 27 May 2017 27 May 2017
Please cite this article as: Kurai, T., Pion wave-function as a bound system in 3+1 dimensional QCD, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.05.028
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Pion wave-function as a bound system in 3+1 dimensional QCD Teruo Kurai Consultant: 1-32-1 Nishifu-machi, Fuchu-shi, Tokyo 183-0031, Japan E-mail:
[email protected] Abstract: We investigate the Bethe-Salpeter-like amplitude in ３＋１ dimensions. This involves a system comprising a quark and an antiquark combined by gauge fields to satisfy gauge invariance. We derive the equation of motion for this system and construct a series of convolution integro-differential equations. By solving these equations and using the Ward Identity, we obtain the pion wave function and then the pion form factor. Our form of the pion form factor explains well the inclination of the large region of
experimental data. Thus, our pion wave function shows the real trailing-off property, which reflects the confinement property. Key words: Bethe-Salpeter-like amplitude, bound system, pion wave function, pion form factor 1. Introduction We showed the result of the ‘t’-Hooft model using a bound system in １ ＋ １ dimensional quantum chromodynamics (QCD) in our previous paper [1]. We obtained consistent results from invoking the gauge invariance between quarks and antiquarks, even though they are in different space positions. In order to obtain the mass spectra, the gauge field string part arose as an important term in a singular integral potential. To investigate further the importance of the gauge field string part, we apply our formulation to obtain the pion wave function and the pion electromagnetic form factor in ３＋１ dimensions. For the pion wave function, E.R. Arriola et al. [2] proposed modified Bessel functions which are consistent with the results of lattice QCD [3][4]. In addition, they showed that the axial vector channel of their wave functions behaved as √
→ ∞, which gave
> 0 as a pion electromagnetic form factor. This
form factor is equivalent to the result proposed by J et al. based on both the RIKEN-BNL-Columbia (RBC) and the United Kingdam’s QCD (UKQCD) collaborations [5]. This form of the pion form factor explains well the behavior of experimental data up 1
to = 0.2 region. For the larger , however, especially for the largest four experimental data, this pion form factor fails to show the inclination of the data. In Section 3, we consider the reason why this discrepancy occurs, but to state simply, the deficit comes from the fact that both J ttner et al. and E.R. Arriola et al. did not consider the confinement potential to obtain their results. While, our form of the pion electromagnetic form factor based on a bound system of ３＋１ dimensional QCD explains well the behavior of both the experimental data in Ref. [16] and the experimental data in Ref. [17] as shown in Fig. 1. 2. Formulation We consider the following hadronic operator, which was proposed by Suura [6], as we worked out the ‘t’Hooft model in Ref. [1]. For a confined system, Suura defined the Bethe-Salpeter-like amplitude as
X 1,2 =< 0| 1,2|/ > ,
(1)
where |0> and |P> denote vacuum state and physical state, respectively and the gauge invariant bilocal operator q 1,2 is defined in the non-Abelian gauge field as
01 1,2 ≡ 34 1 2/56 789 : = 2 from the gauge field string were canceled out by the time derivatives of the quark and antiquark fields. Thus, >= 1 and>= 2 do not exist in the space. This means that we can set >= 1 = >= 2 = 0 . 25
Then, because X=W 5= in our gauge (>V = 0 gauge), X4 .
As >?^ -field is a function of z,
L¨ >?£ = L >?£ = 0 .
In addition, the components of the XV = 0 gauge, we can consider that >? = >?¨ = 0 ¡ → ∞. Thus, d? = dü? = 0 ¡ → ∞.
