Planck scale, Dirac delta function and ultraviolet divergence
Hua Zhang,a,1 Mingshun Yuanb aShenzhen Forms Syntron Information Co.,Ltd., Shenzhen 518000, P.R. China bFujian University of Technology,Mathematics and Physics Institute Fuzhou 350118, P.R. China E-mail: [email protected]
Abstract: The Planck length is the minimum length which physical law do not fail. The Dirac delta function was created to deal with continuous range issue, and it is zero except for one point. Thus contradict the Planck length. Renormalization method is the usual way to deal with divergence difficulties. The authors proposed a new way to solve the problem of ultraviolet divergence and this method is self-consistent with the Planck length.
For this purpose a redefine function δP in position representation was introduced to han- dle with the canonical quantization. The function δP tends to the Dirac delta function if the Planck length goes to zero. By logical deduction the authors obtains new commuta- tion/anticommutation relations and new Feynman propagators which are convergent. This article will deduce the new Feynman propagators for the Klein-Gordon field, the Dirac field and the Maxwell field. Through the new Feynman propagators, we can eliminate ultraviolet divergence. arXiv:1911.01789v5 [physics.gen-ph] 2 Jan 2020
1Corresponding author. Contents
1 Introduction2 1.1 History of renormalization2 1.2 The Planck scale & the Dirac delta function2
2 The renormalization function3 2.1 Rectangular coordinate system3 2.2 Cylindrical coordinate system4 2.3 Spherical coordinate system5
3 Quantization and propagator6 3.1 The Klein-Gordon field6 3.2 The Dirac field8 3.3 The Maxwell field9
4 Feynman loop diagram 10 4.1 Vacuum polarization 11 4.2 Electron self-energy 11
5 Symmetry 12 5.1 The Klein-Gordon field & the Dirac field 12 5.2 Gauge invariance 12
6 Conclusion 13
A Proof of convergence of loop diagram 13 A.1 Rectangular coordinate system 13 A.1.1 Vacuum polarization loop diagram 13 A.1.2 Fermion loop diagram of N photon external lines 15 A.1.3 Electron self-energy loop diagram 16 A.1.4 Arbitrary order case 17 A.2 Cylindrical coordinate system 17 A.2.1 Vacuum polarization loop diagram 17 A.2.2 Electron self-energy loop diagram 18 A.2.3 Arbitrary order case 18
B Calculate the masses of octet baryons from chiral perturbation theory 18 B.1 Rectangular coordinate system 18 B.2 Cylindrical coordinate system 20 B.3 The calculation formulas of mass splitting 20
– 1 – 1 Introduction
1.1 History of renormalization Infinities problem arose when QFT constructed in the late 1920s or in the early 1930s. Many physicists devote to this hard work. If this problem can not be solved, QFT will be abandoned. After World War II, Sinitiro Tomonaga, Julian Schwinger, and Richard Feynman[1–3] found the solution independently, which is called renormalization theory of QED, and it is the most successful theory up to now. The successful calculation of the anomalous magnetic moment of the electron, announce the success of renormalization method. In 1971,Kenneth G. Wilson raised renormalization group[4], which changing cutoff Λ of infinities will not impact the physical calculation results, for the interaction involve one or several momentum-dependent coupling constants. In 1972, Gerard ’t Hooft and Martinus J.G. Veltman proved that non-abelian gauge fields is renormalizable[5]. A criterion[6] for deciding which quantum field theories are renormalizable, that all infinities can be absorbed into a redefinition of a finite number of coupling constants and masses. The goal of this article is, infinities never arise in the procedure of calculation, we can achieve if not count infrared divergence.
1.2 The Planck scale & the Dirac delta function In 1912, Max Planck given the units of length[7], mass, time and temperature from the velocity c of propagation of light in a vacuum, and the constant of gravitation G, and the Planck constant h. In 1999, Peter J. Mohr and Barry N. Taylor published an article list fundamental physical constants[8], with
3 1/2 −35 lP = (~G/c ) = 1.6160(12) × 10 m (1.1) 5 1/2 −44 tP = lP /c = (~G/c ) = 5.3906(40) × 10 s (1.2) In 1927, in order to handle with continuous variables, Paul Dirac introduced a new generalized function[9] which named delta function δ(x). Delta function can be written as[10] 3 δij δ (x − x0) = lim (1.3) ∆Vi→0 ∆Vi Due to the limitation of Planck length, ∆Vi will not tend to zero. A new delta function can be defined as 3 δij δP (x − x0) = (1.4) ∆VP i ∆VP i is the limit case of volume element ∆Vi in Planck length. In different coordinate system, the volume element ∆Vi has different form, thus δP has different form. Our deducing base on the below hypothesis. i. In position representation, the Dirac delta function (not include Kronecker delta func- tion) in canonical commutation/anticommutation relations must redefine as δP function.
δ → δP (1.5)
– 2 – ii. In canonical commutation/anticommutation relations, assume Z Z d3pe±ip.xf(p) → d3pe±ip.x<3(p)f(p) (1.6)
Where f(p) is the function of momentum p. For convenience, we call above item i. and item ii. renormalization transformations. On the action of renormalization transformations, δP function’s Fourier expansion is called renormalization function, denote as <3(p). Renormal- 3 3 ization function < (p) in rectangular coordinate system represent as <⊥(p), in spherical 3 3 coordinate system represent as 2 The renormalization function 2.1 Rectangular coordinate system In the Planck length limit, the particles have determinate place x∈(x 0 − lP /2, x 0 + lP /2), 3 uncertainty ∆x = lP = ∆V . Wave function 3 Ψx0 (x) = δP (x − x 0) (2.1) In this article, if a symbol is added a subscript P or a superscript P , it represents the corresponding symbol under the renormalization transformations. ( 3 3 1/lP , x∈(x 0 − lP /2, x 0 + lP /2) δP (x − x 0) = (2.2) 0, x6∈(x 0 − lP /2, x 0 + lP /2) 3 For δP (x − x 0) its Fourier expansion 1 −ip.x0 3 Ψx0 (p) = p e <⊥(p) (2.3) (2π)3 With 3 lP px lP px lP py lP py lP pz lP pz <⊥(p) = [sin / ][sin / ][sin / ] (2.4) 2~ 2~ 2~ 2~ 2~ 2~ – 3 – l p l p l p l p Figure 1. Renormalization function <2 (p) = [sin P x / P x ][sin P y / P y ] ⊥ 2 2 2 2 ~ ~ ~ 2 ~ , where x-axis and y-axis represent px and py, z-axis represent <⊥(p). 3 3 Compare with Fourier expansion of the delta funtion δ (x − x 0), we got a new item <⊥(p), When lP is infinitesimal, 3 3 lim δP (x − x 0) = δ (x − x 0) (2.5) lP →0 1 1 lim e−ip.x0 <3 (p) = e−ip.x0 (2.6) p 3 ⊥ p 3 lP →0 (2π) (2π) 3 3 We call <⊥(p) as renormalization function in rectangular coordinate system, or call <⊥(p) as momentum distribution wave function in rectangular coordinate system. Compare the renormalization function with Fraunhofer moment aperture diffraction intensity distribution formula, they have similar expression. 2.2 Cylindrical coordinate system 3 πlP In the Planck length limit of cylindrical coordinate system ∆V = 4 , the delta function 4 3 πl3 , r < lP /2, |z| < lP /2 δP cylin(r, θ, z) = P (2.7) 0, r ≥ lP /2, |z| ≥ lP /2 – 4 – Corresponding Fourier expansion 1 −ip.x0 3 Ψx0 (p) = p e Z lP /2 Z 2π Z lP /2 3 4 i[(px cos θ+py sin θ)r+pzz]/~ Where 0F1(a; z) is the generalized hypergeometric series (also see Figure 2./Figure 3./Fig- ure 4., and see [11, 12]), satisfy 2 2 2 ∞ n 2 2 2 lP (px + py) X (−1) lP (px + py) n F1(2; − ) = [ ] (2.11) 0 16 2 n!(n + 1)! 16 2 ~ n=0 ~ 2 2 2 lP (px + py) lim 0F1(2; − 2 ) = 1 (2.12) lP →0 16~ We have 3 lim 2.3 Spherical coordinate system 3 πlP In the Planck length limit of spherical coordinate system ∆V = 6 , the delta function 6 3 πl3 , r < lP /2 δP s(r, θ, ϕ) = P (2.14) 0, r ≥ lP /2 Corresponding Fourier expansion 1 −ip.x0 3 Ψx0 (p) = p e Z 2π Z π Z lP /2 3 6 2 i(px sin ϕ cos θ+py sin ϕ sin θ+pz cos ϕ)r/~ q 2 2 r sin ϕ px + py Z 2π 1 i(px sin ϕ cos θ+py sin ϕ sin θ)r/~ J0( ) = e dθ (2.18) ~ 2π 0 – 5 – 1.0 0.8 0.6 0.4 0.2 -1000 -500 500 1000 -0.2 3 Figure 2. The thick line is the curve of renormalization function We obtain 3 3 lim δP s(r, θ, ϕ) = δ (x − x 0) (2.19) lP →0 1 1 lim e−ip.x0 <3(p) = e−ip.x0 (2.20) p 3 s p 3 lP →0 (2π) (2π) 3 3 Quantization and propagator 3.1 The Klein-Gordon field The usual canonical commutation relations may be written[10, 14] 3 [φP (x, t), πP (y, t)] = iδP (x − y) (3.1) [φP (x, t), φP (y, t)] = 0 (3.2) [πP (x, t), πP (y, t)] = 0 (3.3) – 6 – 3 1 2 2 2 Figure 3. Renormalization function < (p)|p =0 = 0F1(2; − 2 l (p + p )) cylin z 16~ P x y 3 , where x-axis and y-axis represent px and py, z-axis represent (a) (b) 2 1 2 2 lP py lP py Figure 4. Renormalization function – 7 – 3 3 Here with a δP (x − y) in place of a delta function δ (x − y). And Z 3 d k −ik.x † ik.x φP (x, t) = [aP (k)e + a (k)e ] (3.4) p 3 P (2π) 2ωk The commutation relations of annihilate operator and create operator under the renormal- ization transformation, will change to † 0 3 0 3 0 [aP (k), aP (k )] = < (k )δ (k − k ) (3.5) 0 [aP (k), aP (k )] = 0 (3.6) † † 0 [aP (k), aP (k )] = 0 (3.7) Annihilate operator and create operator can be written p 3 aP (k) = < (k)a(k) (3.8) † † p 3 † aP (k) = < (k) a (k) (3.9) Now we can calculate the Feynman propagator from above commutation relations. For the Klein-Gordon field the Feynman propagator may be written(see Figure 5(a)) Z 3 d k −ik.(x−y) 3 ik.(x−y) 3 [i∆F (x, y)]P = 3 [e < (k)Θ(x0 − y0) + e < (k)Θ(y0 − x0)] (2π) 2ωk p 2 2 , ωk = k + m (3.10) Where ( 1, x0 > y0 Θ(x0 − y0) = 0, x0 < y0 The Feynman propagator can be written in a unified form Z 4 3 d k −ik.(x−y) < (k) [i∆F (x, y)]P = i e (3.11) (2π)4 k2 − m2 + i 3.2 The Dirac field After the renormalization transformation the usual canonical anticommutation relations may be written[10] P P 3 [πα (x, t), ψβ (y, t)]+ = iδP (x − y)δαβ (3.12) P P [πα (x, t), πβ (y, t)]+ = 0 (3.13) P P [ψα (x, t), ψβ (y, t)]+ = 0 (3.14) 3 3 Here with a δP (x − y) in place of a delta function δ (x − y). And Z r X 3 m −ip.x † ip.x ψP (x, t) = d p 3 [bP (p, s)u(p, s)e + dP (p, s)v(p, s)e ] (3.15) ±s (2π) ωp – 8 – We can obtain below results from the above canonical anticommutation relations † 0 0 3 3 0 [bP (p, s), bP (p , s )]+ = < (p)δss0 δ (p − p ) (3.16) † 0 0 3 3 0 [dP (p, s), dP (p , s )]+ = < (p)δss0 δ (p − p ) (3.17) [bP , bP ]+ = [dP , dP ]+ = [bP , dP ]+ = 0 (3.18) † † † † † † [bP , bP ]+ = [dP , dP ]+ = [bP , dP ]+ = 0 (3.19) † † [bP , dP ]+ = [bP , dP ]+ = 0 (3.20) Annihilate operator and create operator can be written p 3 bP (k) = < (k)b(k) (3.21) † † p 3 † bP (k) = < (k) b (k) (3.22) p 3 dP (k) = < (k)d(k) (3.23) † † p 3 † dP (k) = < (k) d (k) (3.24) the Feynman propagator(see Figure 5(b)) Z 3 d p −ip.(x−y) 3 ip.(x−y) 3 [iSF (x, y)]P = 3 [e < (p)(m + p/)Θ(x0 − y0) + e < (p)(m − p/) (2π) 2ωp 2 2 2 × Θ(y0 − x0)], ωp = p + m (3.25) it can be written in a unified form Z 4 3 d p −ip.(x−y) < (p) [iSF (x, y)]P = i e (3.26) (2π)4 p/ − m + i 3.3 The Maxwell field In this section the canonical quantization may use radiation gauge. Under the renormal- ization transformations the equal-time commutators are[10] Z 3 P j 3 d k ik.(x−y) 3 kikj [A (x, t), π (y, t)] = iδijδ (x − y) − i e < (k)( ) (3.27) i P P (2π)3 k 2 P P i j [Ai (x, t),Aj (y, t)] = [πP (x, t), πP (y, t)] = 0 (3.28) For the Maxwell field Z 3 2 P d k X −ik.x † ik.x A (x, t) = µ(k, λ)[aP (k, λ)e + a (k, λ)e ] (3.29) µ p 3 P (2π) 2ωk λ=1 where µ(k, λ) = (0, (k, λ)) (3.30) with (k, λ) is polarization vector, satisfy k.(k, λ) = 0, λ = 1, 2 (3.31) 0 (k, λ).(k, λ ) = δλλ0 (3.32) (−k, 1) = −(k, 1), (−k, 2) = +(k, 2) (3.33) – 9 – We can obtain below results for annihilate operator and create operator commutation re- lations † 0 0 3 3 3 0 [aP (k, λ), aP (k , λ )] = [(< (k) + < (−k))/2]δ (k − k )δλλ0 (3.34) 0 0 † † 0 0 [aP (k, λ), aP (k , λ )] = [aP (k, λ), aP (k , λ )] = 0 (3.35) Annihilate operator and create operator can be written r <3(k) + <3(−k) aP (k, λ) = a(k, λ) (3.36) 2 r † <3(k) + <3(−k) a† (k, λ) = a†(k, λ) (3.37) P 2 The photon propagator(see Figure 5(c)) Z 3 2 d k X 3 3 −ik.(x−y) [iDµν(x, y)]P = 3 µ(k, λ)ν(k, λ)[(< (k) + < (−k))/2] × [e Θ(x0 − y0) (2π) 2ωk λ=1 ik.(x−y) + e × Θ(y0 − x0)] (3.38) q √ 2 2 2 with ωk = k + mγ = k . When we set µ µ µ ˆµ k − (k.η)η η = (1, 0, 0, 0), k = p (3.39) (k.η)2 − k2 x y We may obtain 2 ¡2 X kµkν (k.η)(kµην + ηµkν) k ηµην µ(k, λ)ν(k, λ) = −gµν − + − (3.40) (k.η)2 − k2 (k.η)2 − k2 (k.η)2 − k2 λ=1 With gµν is metric in spacetime.Photonx propagator cany be writtenx, α in a unified formy, β 2 Z d4k X e−ik.(x−y) [iD (x, y)] = i (k, λ) (k, λ)[(<3(k) + <3(−k))/2] × (3.41) µν P 4 µ ¡ν ¢2 (2π) k + i λ=1 x y x, α y, β x, µ y, ν (a)¡ Klein-Gordan field ¢(b) Dirac field £(c) Maxwell field Figure 5. Propergators for the Klein-Gordan, Dirac, Maxwell fields 4 Feynman loop diagramx, µ y, ν x, α y, β This section will discuss Feynman loop£ diagrams, the corresponding renormalization func- tion is in¢ rectangular coordinate system or cylindrical coordinate system. The proof that the Feynman integral of loop diagram is convergent refer to appendix. x, µ y, ν £ – 10 – 1 1 1 4.1 Vacuum polarization The Feynman integral of Figure 6(a). Z d4q i<3(q − k) i<3(q) Πµν(k) = (−1)tr[(−ieγµ) (−ieγν) ] (4.1) (2π)4 q/ − k/ − m + i q/ − m + i Above equation can be changed to Z 3 3 3 µν d q µ i(q/ − k/ + m)< (q − k) ν i(q/ + m)< (q) Π (k) = 2 tr[(−ieγ ) (−ieγ ) ] (4.2) 16π ωq−k ωq with p 2 2 ωq−k = (q − k) + m (4.3) p 2 2 ωq = q + m (4.4) Remove the items of equation (4.1) whose trace are zero Z 3 3 3 µν d q µ i(q/ − k/)< (q − k) ν iq/< (q) Π (k) = 2 tr[(−ieγ ) (−ieγ ) ] 16π ωq−k ωq Z 3 3 3 2 d q µ i< (q − k) ν i< (q) + m 2 tr[(−ieγ ) (−ieγ ) ] (4.5) 16π ωq−k ωq Z 3 3 3 2 µ ρ ν λ d q (qρ − kρ)qλ< (q − k) < (q) = e tr[γ γ γ γ ] 2 (4.6) 16π ωq−k ωq Z 3 3 3 2 2 µ ν d q < (q − k) < (q) + m e tr[γ γ ] 2 (4.7) 16π ωq−k ωq 2 µ ρ ν λ 2 2 µ ν = e tr[γ γ γ γ ]Π(ρλ)(k) + m e tr[γ γ ]Π(k) (4.8) 4.2 Electron self-energy The Feynman integral(satisfied Lorenz gauge) of Figure 6(b). Z 4 3 3 2 d k −igµν< (k) µ i< (p − k) ν −iΣ(p/) = (−ie) 4 2 2 γ γ (4.9) (2π) k − mγ + i p/ − k/ − m + i where mγ is the mass of photon.Above equation can be changed to Z 3 3 3 2 d k −igµν< (k) µ i(p/ − k/ + m)< (p − k) ν −iΣ(p/) = e 2 γ γ (4.10) 16π ωk ωp−k with p 2 2 ωp−k = (p − k) + m (4.11) √ 2 ωk = k (4.12) – 11 – (a) Vacuum polarization loop diagram (b) Electron self-energy loop diagram Figure 6. Loop Diagram 5 Symmetry 5.1 The Klein-Gordon field & the Dirac field For equation (3.4), we have 2 ( + m )φP = 0 (5.1) So φP is satisfied the Klein-Gordon equation.The Klein-Gordon equation is Lorentz covari- ance. For equation (3.15), we have µ (iγ ∂µ − m)ψP = 0 (5.2) and ψP is satisfied the Dirac equation. 5.2 Gauge invariance The Lagrangian of scalar field interacting with abelian gauge field † µ 1 µν L = (DµφP ) (D φP ) − [Fµν]P [F ]P − V (φP ) (5.3) 4 with µ µ µ D φP = (∂ − ig[A ]P )φP (5.4) [Fµν]P = [∂µAν − ∂νAµ]P (5.5) where V (φP ) have no derivative of φP . Above Lagrangian is invariance under below trans- formations 0 −igθ(x) φP → φP = e φP (5.6) µ 0µ µ µ [A ]P → [A ]P = [A ]P − ∂ θ (5.7) From the description of this section of quantization and propagator, the renormalization function involve in creation and annihilation operators. Thus above results is obvious. For non-abelian gauge field, it can not use canonical quantization. This paper will not discuss. – 12 – 6 Conclusion In the field of quantum mechanics, the wave function of a particle with definite position is a Dirac function. Because of the limitation of Planck scale, Dirac function needs to be replaced by δP function. It is also used in quantum field theory. The renormalization function involve in creation and annihilation operators, and it is depend on momentum. Under the renormalization transformations, field φP and field ψP satisfy the Klein-Gordon equation and the Dirac equation respectively, and the Lagrangian of scalar field interacting with abelian gauge field is gauge invariant. For free fields, the new Feynman propagators which product a factor of renormalization function(or combination of renormalization function) are convergent in large momentum. For interaction fields, it is simple to prove that the propagators are convergent. The problem of infinities is the result of introducing the Dirac delta function in position representation. When we introduce the Dirac delta function in position representation, this will cause momentum probability amplitude same in any values, and lead to divergence difficulties of large momentum. The introduction of renormalization function can eliminate the ultraviolet divergence, because the limit of large momentum probability amplitude is convergent, and ωk is bounded for large momentum(detail see Appendix), so new Feynman propagators are convergent. A Proof of convergence of loop diagram Proposition[15] (Abel-Dirichlet test for convergence of an integral).Suppose that g is mono- tonic. Then a sufficient condition for convergence of the improper integral Z +∞ f(x)g(x)dx (A.1) a is that the one of the following conditions hold: R +∞ a. (Abel test)the integral a f(x)dx converges, and the function g(x) is bounded on [a, +∞). R A b. (Dirichlet test)the function F (A) = a f(x)dx is bounded on [a, +∞), the function g(x) tends to zero as x → +∞, x∈[a, +∞). R +∞ Theorem[16](Comparison test) ∀x ∈ [a, +∞) if |f(x)| ≤ g(x) and integral a g(x)dx R +∞ is convergent, then integral a f(x)dx is convergent. A.1 Rectangular coordinate system A.1.1 Vacuum polarization loop diagram We will prove that Π(ρλ)(k) and Π(k) in equation (4.8) are convergent. i i 1 Assume qc > k > 0 with k = k = −ki = (kx, ky, kz). So g(q) = is monotonic on ωq−kωq – 13 – 1 [qc, +∞) and g(q) < . By comparison test integral m2 Z +∞ l (qi − ki) l (qi − ki) l qi l qi [sin P / P ][sin P / P ]dqi (A.2) qc 2~ 2~ 2~ 2~ is convergent. By Abel test, integral Z +∞ 3 3 3 d q <⊥(q − k) <⊥(q) 2 (A.3) qc 16π ωq−k ωq is convergent too. Similarly we can prove that integral Z qd 3 3 3 d q <⊥(q − k) <⊥(q) 2 (A.4) −∞ 16π ωq−k ωq i is convergent, with qd < − k < 0. Integral Z qd 3 3 3 d q <⊥(q − k) <⊥(q) 2 (A.5) qc 16π ωq−k ωq is not an improper integral, obviousely it is convergent. Above all integral Z 3 3 3 d q <⊥(q − k) <⊥(q) Π(k) = 2 (A.6) 16π ωq−k ωq is convergent. From above proof, integrals Z +∞ l (qi − ki) l (qi − ki) l qi l qi [sin P / P ][sin P / P ]dqi (A.7) qc 2~ 2~ 2~ 2~ Z qd l (qi − ki) l (qi − ki) l qi l qi [sin P / P ][sin P / P ]dqi (A.8) −∞ 2~ 2~ 2~ 2~ are convergent. Functions s 2 2 (qi − ki) qj g(qi − ki, qj) = (A.9) (q − k)2 + m2 q2 + m2 s 2 qj g(q0 − k0, qj) = (A.10) q2 + m2 s 2 (qi − ki) g(qi − ki, q0) = (A.11) (q − k)2 + m2 are monotonic on [qc, +∞) where qc > |ki| > 0, and g(qi − ki, qj) < 1. By Abel test , integral Z +∞ 3 3 3 d q (qρ − kρ)qλ<⊥(q − k) <⊥(q) 2 (A.12) qc 16π ωq−k ωq – 14 – is convergent. Similarly, integral Z qd 3 3 3 d q (qρ − kρ)qλ<⊥(q − k) <⊥(q) 2 (A.13) −∞ 16π ωq−k ωq is convergent. Proper integral Z qd 3 3 3 d q (qρ − kρ)qλ<⊥(q − k) <⊥(q) 2 (A.14) qc 16π ωq−k ωq is convergent. Above all, improper integral Z +∞ 3 3 3 d q (qρ − kρ)qλ<⊥(q − k) <⊥(q) Π(ρλ)(k) = 2 (A.15) −∞ 16π ωq−k ωq is convergent. So far, Πµν(k) is convergent. (a) N vertices Feynman loop diagram (b) Baryons mesons interaction loop diagram Figure 7. Feynman Diagram A.1.2 Fermion loop diagram of N photon external lines The Feynman integral of Figure 7 (a) (below equation constant factor omitted. We only prove the case which external photon lines are injection momentum , other cases are similar) Z 4 <3 p <3 p − k µ1µ2...µn d p i ⊥( ) µ1 i ⊥( 1) µ2 J1 = 4 (−1)tr{ γ γ ... (2π) p/ − m + i p/ − k/1 − m + i 3 i<⊥(p − k1 − ... − kn−1) 4 × }δ (k1 + k2 + ... + kn) (A.16) p/ − k/1 − ... − k/n−1 − m + i Z d3p = (−1)(π)ntr{(p/ + m)γµ1 (p/ − k/ + m)γµ2 ...(p/ − k/ − ... − k/ + m)} (2π)4 1 1 n−1 3 3 3 <⊥(p) <⊥(p − k1) <⊥(p − k1 − ... − kn−1) 4 × ... δ (k1 + k2 + ... + kn) (A.17) ωp ωp−k1 ωp−k1−...−kn−1 with p 2 2 ωp = p + m (A.18) p 2 2 ωp−k1 = (p − k1) + m (A.19) ... – 15 – If we can prove integral Z 3 3 3 3 d p <⊥(p) <⊥(p − k1) <⊥(p − k1 − ... − kn−1) 3 ... (2π) ωp ωp−k1 ωp−k1−...−kn−1 × pν1 (pν2 − k1ν2 )...(pνn − k1νn ... − k(n−1)νn ) (A.20) is convergent, then equation (A.16) is convergent. Function s 2 2 − − − 2 pi (pi1 − k1i1 ) (pin−1 k1in−1 ... k(n−1)in−1 ) g(pi, pi1 ..., pin−1 ) = 2 2 2 2 ... 2 2 p + m (p − k1) + m (p − k1 − ... − kn−1) + m (A.21) is monotonic on pi ∈ [qc, +∞) where qc > |k1i|+|k2i|+...+ k(n−1)i > 0, and g(pi, pi1 ..., pin−1 ) < 1. Similar handle procedure with (A.7) (A.8) , we can obtain integrals Z +∞ l pi l pi l (pi − ki ) l (pi − ki ) [sin P / P ][sin P 1 / P 1 ]... qc 2~ 2~ 2~ 2~ i i i i i i lP (p − k − ... − k ) lP (p − k − ... − k ) × [sin 1 n−1 / 1 n−1 ]dpi (A.22) 2~ 2~ Z qd l pi l pi l (pi − ki ) l (pi − ki ) [sin P / P ][sin P 1 / P 1 ]... −∞ 2~ 2~ 2~ 2~ i i i i i i lP (p − k − ... − k ) lP (p − k − ... − k ) × [sin 1 n−1 / 1 n−1 ]dpi (A.23) 2~ 2~ i i i are convergent(where qd < − k1 − k2 − ... − kn−1 < 0). By Abel test, integral Z +∞ 3 3 3 3 d p <⊥(p) <⊥(p − k1) <⊥(p − k1 − ... − kn−1) 3 ... qc (2π) ωp ωp−k1 ωp−k1−...−kn−1 × pν1 (pν2 − k1ν2 )...(pνn − k1νn ... − k(n−1)νn ) (A.24) is convergent. Similarly, integral Z qd 3 3 3 3 d p <⊥(p) <⊥(p − k1) <⊥(p − k1 − ... − kn−1) 3 ... −∞ (2π) ωp ωp−k1 ωp−k1−...−kn−1 × pν1 (pν2 − k1ν2 )...(pνn − k1νn ... − k(n−1)νn ) (A.25) convergent. We obtain integral (A.20) is convergent. Thus equation (A.16) is convergent. A.1.3 Electron self-energy loop diagram From equation (4.10), we have Z 3 3 3 2 d k <⊥(k) µ (p/ − k/ + m)<⊥(p − k) ν −iΣ(p/) = e gµν 2 γ γ (A.26) 16π ωk ωp−k Z 3 3 3 2 µ ρ ν d k <⊥(k) (pρ − kρ)<⊥(p − k) = e gµνγ γ γ 2 (A.27) 16π ωk ωp−k Z 3 3 3 2 µ ν d k <⊥(k) <⊥(p − k) + e mgµνγ γ 2 (A.28) 16π ωk ωp−k Compare (A.27) (A.28) with (4.6)(4.7) and from above proof, we can obtain −iΣ(p/) is convergent on large momentum. – 16 – A.1.4 Arbitrary order case Consider integral 3 3 3 Z < (p − k1) < (p − k2) < (p − kn) d3p ⊥ ⊥ ... ⊥ ωp−k1 ωp−k2 ωp−kn × (pν1 − k1ν1 )(pν2 − k2ν2 )...(pνn − knνn ) (A.29) with q 2 2 ωp−kj = (p − kj) + mj (A.30) j = 1, 2, ..., n From proof procedure of (A.20), we obtain above integral is convergent. A.2 Cylindrical coordinate system A.2.1 Vacuum polarization loop diagram Firstly we may proove that integral Z +∞ Z +∞ Z +∞ 2 2 2 lP (px + py) lP pz lP pz 0F1(2; − 2 )[sin / ]dpxdpydpz (A.31) −∞ −∞ −∞ 16~ 2~ 2~ 2 2 2 lP (px+py) is convergent. From Cauchy-Hadamard theorem[16], we obtain power series F1(2; − 2 ) 0 16~ is convergent for any px and py. By calculate Z +∞ Z +∞ 2 2 2 lP (px + py) 0F1(2; − 2 )dpxdpy −∞ −∞ 16~ ∞ Z +∞ Z 2π X (−1)n l2 r2 = r ( P )ndrdθ (A.32) n!(n + 1)! 16 2 0 0 n=0 ~ 2 ∞ n 16π~ X (−1) n+1 = lim r (A.33) l2 r→+∞ [(n + 1)!]2 P n=0 2 16π~ = 2 lP And integral Z +∞ lP pz lP pz sin / dpz (A.34) −∞ 2~ 2~ is convergent. Thus (A.31) is convergent. We can prove that integral Z +∞ 3 3 3 d q – 17 – 2 2 lP r is convergent. As power series F1(2; − 2 ) is convergent for any r ∈ Real, and 0 16~ 2 2 lP r lim 0F1(2; − ) = 0 (A.36) r→+∞ 16~2 2 2 lP r Assume F1(2; − 2 ) ≤ Fmax. From Cauchy-Hadamard theorem, obtain integral 0 16~ Z +∞ 3 3 3 d q A.2.2 Electron self-energy loop diagram The process of proof similar with case in the previous section which use Cauchy-Hadamard theorem and Abel-Dirichlet test. A.2.3 Arbitrary order case Consider integral Z 3 3 3 × (pν1 − k1ν1 )(pν2 − k2ν2 )...(pνn − knνn ) (A.38) with q 2 2 ωp−kj = (p − kj) + mj (A.39) j = 1, 2, ..., n By Cauchy-Hadamard theorem, integral Z 3 3 3 3 d p B Calculate the masses of octet baryons from chiral perturbation theory B.1 Rectangular coordinate system In this section,the authors will calculate one loop graph from renormalization function 3 <⊥(k), and calculate the masses of octet baryons from chiral perturbation theory[17, 18]. The Figure 7 (b) loop graph contribution Z µ 1 3 σ[M1,M2] = −γ γ5 d k(m + p/ − k/)(kµ − ∆µ)kν 16π 3 3 3 <⊥(p − k)<⊥(k)<⊥(k − ∆) ν × γ γ5 (B.1) ωp−kωkωk−∆ – 18 – with q 2 2 ωp−k = (p − k) + m (B.2) q 2 2 ωk = k + M1 (B.3) q 2 2 ωk−∆ = (k − ∆) + M2 (B.4) Octet baryons mean mass m = 1151MeV (B.5) Pseudoscalar mesons experimental values of masses Mπ0 = 134.96MeV, Mη = 548.8MeV, Mπ+ = Mπ− = 139.57MeV, MK+ = MK− = 493.71MeV 0 0 MK = MK = 497.70MeV The calculation results for loop graph σ[Mη,Mη] = −577.2830, σ[Mπ0 ,Mη] = −574.3426, σ[Mπ0 ,Mπ0 ] = −565.3559, σ[Mπ+ ,Mπ− ] = −566.1214, + − − 0 0 − (B.6) σ[MK ,MK ] = 577.0600, σ[MK ,MK ] = 577.0778 Quark experimental values of masses mu = 2MeV, md = 5MeV, ms = 150MeV (B.7) Constants gA = 1.25,B = 2.1MeV, fπ = 93.3MeV (B.8) Here assume constant B absorb the constant from calculating loop graph contribution σ[M1,M2] (if B keep unchanged, then need add a new constant gRF , satisfied B∗gRF = 2.1, thus keep calculation results unchanged). Set cr = 547.8883, d = 0.6983 (B.9) Which make below equation 8 X cal exp ∆mRF = Abs(mB − mB ) (B.10) B=1 exp a minimum value (∆mRF = 58.78).With mB represent the experimental values of oectet cal baryons masses, and mB represent the theoretical calculation values. From above param- eter setting, we can obtain the theoretical calculation results(the calculation formulas of mass splitting refer to citation[17]) mp = 938.28MeV, mn = 940.31MeV, mΛ = 1092.02MeV, mΣ0 = 1199.09MeV, mΣ+ = 1178.02MeV, mΣ− = 1183.62MeV, mΞ0 = 1317.72MeV, mΞ− = 1321.30MeV (B.11) – 19 – B.2 Cylindrical coordinate system In this section, the authors will calculate the results in cylindrical coordinate system. The Figure 7 (b) loop graph contribution Z µ 1 3 σ[M1,M2] = −γ γ5 d k(m + p/ − k/)(kµ − ∆µ)kν 16π <3 p − k <3 k <3 k − ∆ cylin( ) cylin( ) cylin( ) ν × γ γ5 (B.12) ωp−kωkωk−∆ By integrate highly ocillatory functions, we obtain calculation results for loop graph σ[Mη,Mη] = −765.8240, σ[Mπ0 ,Mη] = −760.9211, σ[Mπ0 ,Mπ0 ] = −745.9441, σ[Mπ+ ,Mπ− ] = −747.2172, + − − 0 0 − (B.13) σ[MK ,MK ] = 765.4512, σ[MK ,MK ] = 765.4819 By similar methods, set cr = 0.7156, d = 0.6862,B = 0.12 (B.14) and obtain ∆mRF = 59.565 (B.15) it is a minimum value. Input these parameters to the calculation formulas of mass splitting mp = 938.02MeV, mn = 939.93MeV, mΛ = 1086.98MeV, mΣ0 = 1208.07MeV, mΣ+ = 1184.57MeV, mΣ− = 1190.12MeV, mΞ0 = 1317.65MeV, mΞ− = 1321.30MeV (B.16) B.3 The calculation formulas of mass splitting Here we list the calculation formulas of mass splitting, detail see[17] 2 2 2 Bm crκ2 gAκ2 d 2dgA gA √ { − − 0 0 − 0 ∆mp = 2 c3[ σ[MK ,MK ] + ( + ) κ4σ[Mπ ,Mη] fπ 3 4 3 3 4 2 2 2 2d gA 5crκ2 2crκ4 d dgA + ( − dgA + )κ2σ[MK+ ,MK− ]] + c8[− √ − √ + ( √ − √ 3 2 3 3 3 3 2 3 2 3 √ 2 2 2 2 gA d dgA 3gA gA + √ )σ[Mπ0 ,Mπ0 ] + (− √ + √ − )κ4σ[Mη,Mη] − √ κ2σ[MK0 ,M 0 ] 8 3 6 3 2 3 8 4 3 K 2 2 2 2 2d dgA gA 2d 2dgA gA + (− √ + √ − √ )κ2σ[MK+ ,MK− ] + ( √ − √ + √ )σ[Mπ+ ,Mπ− ]]} 3 3 3 2 3 3 3 2 3 (B.17) – 20 – 2 2 2 2 Bm crκ2 2d gA d 2dgA gA √ { − − 0 0 − − ∆mn = 2 c3[ ( dgA + )κ2σ[MK ,MK ] ( + ) κ4 fπ 3 3 2 3 3 4 2 2 gA 5crκ2 2crκ4 d dgA × σ[Mπ0 ,Mη] + κ2σ[MK+ ,MK− ]] + c8[− √ − √ + ( √ − √ 4 3 3 3 3 2 3 2 3 √ 2 2 2 2 2 gA d dgA 3gA 2d dgA gA + √ )σ[Mπ0 ,Mπ0 ] + (− √ + √ − )κ4σ[Mη,Mη] + (− √ + √ − √ ) 8 3 6 3 2 3 8 3 3 3 2 3 2 2 2 gA 2d 2dgA gA × κ2σ[MK0 ,M 0 ] − √ κ2σ[MK+ ,MK− ] + ( √ − √ + √ )σ[Mπ+ ,Mπ− ]]} K 4 3 3 3 2 3 (B.18) 2 2 2 2 Bm 5d 3dgA 3gA 5d 3dgA 3gA { − − 0 0 − ∆mΛ = 2 c3[( + )κ2σ[MK ,MK ] + ( + )κ2 fπ 6 2 4 6 2 4 2 2 5cr 5crκ2 crκ4 d d × σ[MK+ ,MK− ]] + c8[ √ − √ − √ + √ σ[Mπ0 ,Mπ0 ] − √ κ4σ[Mη,Mη] 6 3 3 3 6 3 6 3 6 3 √ √ 2 2 2 5d 3dgA 3gA 2d + (− √ + − )κ2(σ[MK0 ,M 0 ] + σ[MK+ ,MK− ]) + √ σ[Mπ+ ,Mπ− ]]} 6 3 2 4 K 3 3 (B.19) 2 2 2 2 Bm d dgA gA d dgA gA 0 { − − 0 0 − ∆mΣ = 2 c3[( + )κ2σ[MK ,MK ] + ( + )κ2 fπ 2 2 4 2 2 4 2 2 13cr crκ2 crκ4 d d × σ[MK+ ,MK− ]] + c8[ √ − √ − √ + √ σ[Mπ0 ,Mπ0 ] − √ κ4σ[Mη,Mη] 6 3 3 6 3 6 3 6 3 2 2 d dgA gA + (− √ + √ − √ )κ2(σ[MK0 ,M 0 ] + σ[MK+ ,MK− ]) 2 3 2 3 4 3 K 2 2 2d 4dgA 2gA + ( √ − √ + √ )σ[Mπ+ ,Mπ− ]]} (B.20) 3 3 3 √ 2 2 Bm cr κ4 gA d dgA √ + { − − − 0 0 − 0 ∆mΣ = 2 c3[ crκ2 κ2σ[MK ,MK ] + ( ) κ4σ[Mπ ,Mη] fπ 3 4 3 3 2 2 2 2 gA 11cr crκ2 crκ4 d dgA gA + (d − dgA + )κ2σ[MK+ ,MK− ]] + c8[ √ − √ − √ + ( √ − √ + √ ) 4 6 3 3 6 3 2 3 3 2 3 2 2 2 2 d gA d dgA gA × σ[Mπ0 ,Mπ0 ] − √ κ4σ[Mη,Mη] − √ κ2σ[MK0 ,M 0 ] + (−√ + √ − √ )κ2 6 3 4 3 K 3 3 4 3 2 2 4d 2dgA gA × σ[MK+ ,MK− ] + ( √ − √ + √ )σ[Mπ+ ,Mπ− ]]} (B.21) 3 3 3 3 – 21 – √ 2 2 Bm cr κ4 2 gA d dgA √ − { − − 0 0 − ∆mΣ = 2 c3[crκ2 + + ( d + dgA )κ2σ[MK ,MK ] + ( + ) κ4 fπ 3 4 3 3 2 2 2 gA 11cr crκ2 crκ4 d dgA gA × σ[Mπ0 ,Mη] + κ2σ[MK+ ,MK− ]] + c8[ √ − √ − √ + ( √ − √ + √ ) 4 6 3 3 6 3 2 3 3 2 3 2 2 2 2 d d dgA gA gA × σ[Mπ0 ,Mπ0 ] − √ κ4σ[Mη,Mη] + (−√ + √ − √ )κ2σ[MK0 ,M 0 ] − √ κ2 6 3 3 3 4 3 K 4 3 2 2 4d 2dgA gA × σ[MK+ ,MK− ] + ( √ − √ + √ )σ[Mπ+ ,Mπ− ]]} (B.22) 3 3 3 3 √ 2 2 2 Bm 2crκ2 cr κ4 2d gA d dgA √ 0 { − − − − 0 0 − ∆mΞ = 2 c3[ + ( + dgA )κ2σ[MK ,MK ] + ( ) κ4 fπ 3 3 3 2 3 4 2 2 2 gA 5cr 2crκ2 crκ4 gA × σ[Mπ0 ,Mη] + (d − dgA + )κ2σ[MK+ ,MK− ]] + c8[ √ − √ − √ + √ 4 2 3 3 3 6 3 8 3 √ 2 2 2 2 2d dgA 3gA 2d dgA gA × σ[Mπ0 ,Mπ0 ] + (− √ + √ − )κ4σ[Mη,Mη] + (− √ + √ − √ )κ2 3 3 3 8 3 3 3 2 3 2 2 2 d dgA gA gA × σ[MK0 ,M 0 ] + (−√ + √ − √ )κ2σ[MK+ ,MK− ] + √ σ[Mπ+ ,Mπ− ]]} K 3 3 4 3 2 3 (B.23) √ 2 2 Bm 2crκ2 cr κ4 2 gA d dgA √ − { − − 0 0 − ∆mΞ = 2 c3[ + + ( d + dgA )κ2σ[MK ,MK ] + ( + ) κ4 fπ 3 3 4 3 4 2 2 2 2d gA 5cr 2crκ2 crκ4 gA × σ[Mπ0 ,Mη] + ( − dgA + )κ2σ[MK+ ,MK− ]] + c8[ √ − √ − √ + √ 3 2 2 3 3 3 6 3 8 3 √ 2 2 2 2 2d dgA 3gA d dgA gA × σ[Mπ0 ,Mπ0 ] + (− √ + √ − )κ4σ[Mη,Mη] + (−√ + √ − √ )κ2 3 3 3 8 3 3 4 3 2 2 2 2d dgA gA gA × σ[MK0 ,M 0 ] + (− √ + √ − √ )κ2σ[MK+ ,MK− ] + √ σ[Mπ+ ,Mπ− ]]} K 3 3 3 2 3 2 3 (B.24) References [1] R. 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