Finite Quantum Field Theory and Renormalization Group M. A. Greena and J. W. Moffata;b aPerimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada bDepartment of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada September 15, 2021 Abstract Renormalization group methods are applied to a scalar field within a finite, nonlocal quantum field theory formulated perturbatively in Euclidean momentum space. It is demonstrated that the triviality problem in scalar field theory, the Higgs boson mass hierarchy problem and the stability of the vacuum do not arise as issues in the theory. The scalar Higgs field has no Landau pole. 1 Introduction An alternative version of the Standard Model (SM), constructed using an ultraviolet finite quantum field theory with nonlocal field operators, was investigated in previous work [1]. In place of Dirac delta-functions, δ(x), the theory uses distributions (x) based on finite width Gaussians. The Poincar´eand gauge invariant model adapts perturbative quantumE field theory (QFT), with a finite renormalization, to yield finite quantum loops. For the weak interactions, SU(2) U(1) is treated as an ab initio broken symmetry group with non- zero masses for the W and Z intermediate× vector bosons and for left and right quarks and leptons. The model guarantees the stability of the vacuum. Two energy scales, ΛM and ΛH , were introduced; the rate of asymptotic vanishing of all coupling strengths at vertices not involving the Higgs boson is controlled by ΛM , while ΛH controls the vanishing of couplings to the Higgs. Experimental tests of the model, using future linear or circular colliders, were proposed. Present observations are consistent with ΛM 10 TeV. ≥ The Higgs boson mass hierarchy problem will be solved if future experiments confirm the prediction ΛH . 1 TeV. In the following, we will investigate the consequences of an application of renormalization group (RG) methods for the perturbative finite renormalizable model. We will concentrate on a nonlocal spin 0 scalar field φ = φH Lagrangian model which is perturbatively formulated in Euclidean momentum space and might describe the Higgs boson field if nonlocality were fundamental. The ultraviolet finite theory resolves the Higgs mass hierarchy problem, the scalar field model triviality problem and removes the Landau pole singularity for the Higgs field. 2 Scalar Field Theory arXiv:2012.04487v2 [physics.gen-ph] 14 Sep 2021 The Lagrangian we consider for a real scalar field φ φH describing the Higgs boson in Euclidean space is ≡ 1 2 2 1 4 H = ( φ φ + m φ ) + λ0φ : (1) L 2 − 0 4! Using the formalism of [3], we assume that the vacuum expectation of the bare field φ vanishes and 1=2 write φ = Z φr, where φr is the renormalized field. Expressed as series expansions in powers of the physical coupling λ, mass m and energy scale ΛH , the field strength renormalization constant Z and the bare parameters m0 and λ0 are given by: 2 Z = 1 + δZ(λ, m; ΛH ); (2) 1 2 2 2 2 Zm0 = m + δm (λ, m; ΛH ); (3) 2 2 Z λ0 = λ + δλ(λ, m; ΛH ): (4) The propagator in Euclidean momentum space is given by i 2(p) i∆H (p) E ; (5) ≡ p2 + m2 where (p) is the entire function: E p2 + m2 (p) = exp 2 : (6) E − 2ΛH Evaluating the one-loop self-energy graph gives a constant shift to the Higgs boson bare self-energy [3]: −2 2 iZ λ 2 m iΣ0 = − 2 m Γ 1; 2 ; (7) − 32π − ΛH where Γ(n; z) is the incomplete gamma function: Z 1 Γ(n; z) = dt tn−1 exp( t) = (n 1)Γ(n 1; z) + zn−1 exp( z): (8) z − − − − Setting n = 0 in (8) gives: 1 Z 1 exp( t) X ( z)n Γ(0; z) = E1(z) = dt − = ln(z) γ − ; (9) t nn! z − − − n=1 exp( z) Γ( 1; z) = Γ(0; z) + − : (10) − − z 2 The renormalized one-loop self-energy ΣR(p ) can then be written in the form: −1 2 2 2 2 2 Z λ 2 m 2 ΣR(p ) = δZ(p + m ) + δm + 2 m Γ 1; 2 + (λ ): (11) 32π − ΛH O The renormalized mass and field strength are given by 2 2 λ 2 m 2 δm = 2 m Γ 1; + (λ ); (12) −32π − ΛH O δZ = (λ2): (13) O The expansion of the one-loop Higgs boson self-energy mass correction for m ΛH is 2 2 2 λ 2 2 ΛH 2 m 2 δm = 2 ΛH + m ln 2 + m (1 γ) + 2 + (λ ): (14) 32π − m − O ΛH O The one-loop vertex correction is given by 3λ2 Z 1=2 1 m2 δλ = dx Γ 0; + (λ3): (15) 16π2 1 x Λ2 O 0 − H For m ΛH this can be expanded for the Higgs boson to give 2 2 2 3λ 1 ΛH 1 m 3 δλ = 2 ln 2 + (ln(2) 1 γ) + 2 + (λ ): (16) 16π 2 m 2 − − O ΛH O 2 3 Callan-Symanzik Equation and Running of λ Let us consider the Callan-Symanzik equations [13, 14, 15, 16] satisfied with our energy (length) scales Λi playing the roles of finite renormalization scales. In finite QFT theory, the equations for the regularized amplitudes Γ(n)(x x0) are − @ @ (n) Λi + β(gi) 2γ(gi) Γ = 0; (17) @Λi @gi − where gi are the running coupling constants associated with diagram vertices. The correlation functions (n) will satisfy this equation for the n-th order Γ for the Gell-Mann-Low functions β(gi) and the anomalous dimensions in nth-loop order. For the Higgs field, the RG equation is given by @ @ H ΛH + β(λ) 2γ(λ) Γ = 0: (18) @ΛH @λ − where the coupling λ runs with ΛH . Neglecting the anomalous dimension term γ(λ) and replacing the measured Higgs mass m by the RG scaling mass µ yields the equation: dλ β(λ) = : (19) − ΛH d ln µ We obtain from (15) the Higgs field β function: 3λ2 β(λ) = I(µ2=Λ2 ) + (λ3); (20) 16π2 H O where Z 1=2 1 µ2 I(µ2=Λ2 ) = dx Γ 0; : (21) H 1 x Λ2 0 − H Using the identities Γ(0; y) = E1(y) = Ei( y), yields: − − Z 1=2 2 2 2 1 µ 1 −2µ2 2µ2 −2µ2 I(µ =ΛH ) = dx Ei 2 = 2 exp Λ2 + 1 + Λ2 Ei Λ2 − −1 x Λ H H H 0 − H −µ2 µ2 −µ2 exp Λ2 1 + Λ2 Ei Λ2 : (22) − H − H H We have λ = λ0 + δλ and dλ dδλ = = β(λ): (23) ΛH ΛH − d µ d ln µ From (20) we obtain dλ 3 = dI(µ2=Λ2 ): (24) λ2 −16π2 H Integrating this equation we get 1 1 2 2 = + J(µ =ΛH ); (25) λ λ0 where Z 2 2 3 dΛH 2 2 J(µ =ΛH ) = 2 I(µ =ΛH ): (26) 16π ΛH 2 2 µ2 Evaluating the integral for J(µ =ΛH ), using x = 2 , gives ΛH 3 J(x) = 2 exp( 2x) + 4 exp( x) + π2 (2 + 4x)Ei( 2x) + (4 + 4x)Ei( x) 128π2 − − − − − − +4x 3F3(1; 1; 1; 2; 2; 2; 2x) 4x 3F3(1; 1; 1; 2; 2; 2; x) − − − ln(2)2 ln(4)γ + 2(γ ln(2)) ln(x) + ln(x)2 ; (27) − − − 3 where pFq(a1; :::; ap; b1; :::; bq; z) is a generalized hypergeometric function. From (25) we obtain: λ0 λ = 2 2 ; (28) 1 + λ0J(µ =ΛH ) or λ λ0 = : (29) 1 λJ(µ2=Λ2 ) − H We can compare (25) with the equation obtained in SM: 1 1 3 ΛC = + 2 ln ; (30) λ λ0 16π µ or λ0 λ = ; (31) 3λ0 ΛC 1 + 16π2 ln µ and λ λ0 = : (32) 3λ ΛC 1 2 ln − 16π µ In the SM, the λφ4 model is renormalizable and produces finite scattering amplitudes and cross sections, but renormalization theory demands that the cutoff ΛC must be taken to infinity, ΛC [5, 6]. Then, from (30), the renormalized coupling constant λ = 0. This is known as the triviality problem! 1 [7, 8, 9, 10, 11, 12]. This result holds even in the limit λ0 : ! 1 1 3 Λ ln C : (33) λ ∼ 16π2 µ In the earlier paper [4], it was demonstrated that the triviality problem for the scalar field field could be resolved in the finite QFT theory. Because ΛH = 1=`H is a fundamental constant to be measured, it cannot be taken to infinity as in the case of infinite renormalization theory. Thus, we cannot take the limit `H 0 ! corresponding to the δ-function limit. From Fig. 1, we observe that when we choose ΛH . 1 TeV, the Higgs mass hierarchy problem is resolved, for we have δm2=m2 (1) where m = 125 GeV. From Fig. 1, we 1 ∼ O observe that for ΛH > 2 µ, we avoid a Landau pole and, in particular, for 700 < ΛH < 1 TeV, we resolve the triviality problem for the scalar Higgs field and the Higgs mass fine-tuning hierarchy problem. Choosing an energy µ0 above ΛH as a measurement probe of the running of λ is attempting to make a measurement within the finite Gaussian distribution length size `H [1] and is prohibited within the pertur- bation approximations we have assumed. The results obtained for the running of λ are for a single Higgs particle interacting with another Higgs particle. This cannot describe a fully realistic situation, for the Higgs coupling to other particles such as the top quark (the top quark-Higgs coupling λt (1)) may play an important role. ∼ O The above one-loop calculations have employed a perturbative formulation in Euclidean momentum space and rely on analytic continuation to obtain corresponding Lorentzian results.