Finite Quantum Field Theory and Renormalization Group
FFinite Quantum Field Theory and Renormalization Group
M. A. Green a and J. W. Moffat a,b a Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada b Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
December 9, 2020
Abstract
Renormalization group methods are applied to the scalar field theory of a finite quantum field theory. Itis demonstrated that the triviality problem in scalar field theory, the Higgs boson mass hierarchy problemand the stability of the vacuum are resolved in the theory. The scalar Higgs field has no Landau pole.
An alternative version of the Standard Model (SM), constructed using an ultraviolet finite quantum fieldtheory with nonlocal field operators, was investigated in previous work [1]. In place of Dirac delta-functions, δ ( x ), the theory uses distributions E ( x ) based on finite width Gaussians. The Poincar´e and gauge invariantmodel adapts perturbative quantum field theory (QFT), with a finite renormalization, to yield finite quantumloops. 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. Themodel guarantees the stability of the vacuum. Two energy scales, Λ M and Λ H , were introduced; the rateof asymptotic vanishing of all coupling strengths at vertices not involving the Higgs boson is controlled byΛ M , while Λ H controls the vanishing of couplings to the Higgs. Experimental tests of the model, usingfuture linear or circular colliders, were proposed. Present observations are consistent with Λ M ≥
10 TeV.The Higgs boson mass hierarchy problem will be solved if future experiments confirm the prediction Λ H (cid:46) φ = φ H Lagrangian model describing the Higgs boson field. The ultraviolet finite theory resolves the Higgsmass hierarchy problem, the scalar field model triviality problem and removes the Landau pole singularityfor the Higgs field.
The Lagrangian we consider for a real scalar field φ ≡ φ H describing the Higgs boson in Euclidean space is L H = 12 ( − φ (cid:3) φ + m φ ) + 14! λ φ . (1)Using the formalism of [3], we assume that the vacuum expectation of the bare field φ vanishes andwrite φ = Z / φ r , where φ r is the renormalized field. Expressed as series expansions in powers of thephysical coupling λ , mass m and energy scale Λ H , the field strength renormalization constant Z and thebare parameters m and λ are given by: Z = 1 + δZ ( λ, m, Λ H ) , (2) Zm = m + δm ( λ, m, Λ H ) , (3)1 a r X i v : . [ phy s i c s . g e n - ph ] D ec λ = λ + δλ ( λ, m, Λ H ) . (4)The propagator in Euclidean momentum space is given by i ∆ H ( p ) ≡ i E ( p ) p + m , (5)where E ( p ) is the entire function: E ( p ) = exp (cid:20) − (cid:18) p + m H (cid:19)(cid:21) . (6)Evaluating the one-loop self-energy graph gives a constant shift to the Higgs boson bare self-energy [3]: − i Σ = − iZ − λ π m Γ (cid:18) − , m Λ H (cid:19) , (7)where Γ( n, z ) is the incomplete gamma function:Γ( n, z ) = (cid:90) ∞ z dt t n − exp( − t ) = ( n − n − , z ) + z n − exp( − z ) . (8)Setting n = 0 in (8) gives:Γ(0 , z ) = E ( z ) = (cid:90) ∞ z dt exp( − t ) t = − ln( z ) − γ − ∞ (cid:88) n =1 ( − z ) n nn ! , (9)Γ( − , z ) = − Γ(0 , z ) + exp( − z ) z . (10)The renormalized one-loop self-energy Σ R ( p ) can then be written in the form:Σ R ( p ) = δZ ( p + m ) + δm + Z − λ π m Γ (cid:18) − , m Λ H (cid:19) + O ( λ ) , (11) δZ = O ( λ ) . (12)The expansion of the one-loop Higgs boson mass correction is δm = λ π (cid:20) − Λ H + m ln (cid:18) Λ H m (cid:19) + m (1 − γ ) + O (cid:18) m Λ H (cid:19)(cid:21) + O ( λ ) . (13)The one-loop vertex correction is given by δλ = 3 λ π (cid:90) / dx Γ (cid:18) , − x m Λ H (cid:19) + O ( λ ) . (14)For m (cid:28) Λ H this can be expanded to give δλ = 3 λ π (cid:20) ln (cid:18) Λ H m (cid:19) + 12 (ln(2) − − γ ) + O (cid:18) m Λ H (cid:19)(cid:21) + O ( λ ) . (15) λ Let us consider the Callan-Symanzik equations [12, 13, 14, 15] satisfied with our energy (length) scales Λ i playing the roles of finite renormalization scales. In finite QFT theory, the equations for the regularizedamplitudes Γ ( n ) ( x − x (cid:48) ) are (cid:20) Λ i ∂∂ Λ i + β ( g i ) ∂∂g i − γ ( g i ) (cid:21) Γ ( n ) = 0 , (16)2here g i are the running coupling constants associated with diagram vertices. The correlation functionswill satisfy this equation for the n-th order Γ ( n ) for the Gell-Mann-Low functions β ( g i ) and the anomalousdimensions in nth-loop order.For the Higgs field, the RG equation is given by (cid:20) Λ H ∂∂ Λ H + β ( λ ) ∂∂λ − γ ( λ ) (cid:21) Γ H = 0 . (17)where the coupling λ runs with Λ H . Neglecting the anomalous dimension term γ ( λ ) and replacing themeasured Higgs mass m H by the RG scaling mass µ yields the equation: β ( λ ) = − dλd ln (cid:16) Λ H µ (cid:17) . (18)We obtain from (14) the Higgs field β function: β ( λ ) = 3 λ π I ( µ / Λ H ) + O ( λ ) , (19)where I ( µ / Λ H ) = (cid:90) / dx Γ (cid:18) , − x µ Λ H (cid:19) . (20)Using the identities Γ(0 , y ) = E ( y ) = − Ei( − y ), yields: I ( µ / Λ H ) = − (cid:90) / dx Ei (cid:18) − − x µ Λ H (cid:19) = (cid:16) exp (cid:16) − µ Λ H (cid:17) + (cid:16) µ Λ H (cid:17) Ei (cid:16) − µ Λ H (cid:17)(cid:17) − exp (cid:16) − µ Λ H (cid:17) − (cid:16) µ Λ H (cid:17) Ei (cid:16) − µ Λ H (cid:17) . (21)We have λ = λ + δλ and dλd (cid:16) Λ H µ (cid:17) = dδλd ln (cid:16) Λ H µ (cid:17) = − β ( λ ) . (22)From (19) we obtain dλλ = − π dI ( µ / Λ H ) . (23)Integrating this equation we get 1 λ = 1 λ + J ( µ / Λ H ) , (24)where J ( µ / Λ H ) = 316 π (cid:90) d Λ H Λ H I ( µ / Λ H ) . (25)Evaluating the integral for J ( µ / Λ H ), using x = µ Λ H , gives J ( x ) = 3128 π (cid:0) − − x ) + 4 exp( − x ) + π − (2 + 4 x )Ei( − x ) + (4 + 4 x )Ei( − x )+4 x F (1 , ,
1; 2 , , − x ) − x F (1 , ,
1; 2 , , − x ) − ln(2) − ln(4) γ + 2( γ − ln(2)) ln( x ) + ln( x ) (cid:1) , (26)where p F q ( a , ..., a p ; b , ..., b q ; z ) is a generalized hypergeometric function.From (24) we obtain: λ = λ λ J ( µ / Λ H ) , (27)3r λ = λ − λJ ( µ / Λ H ) . (28)We can compare (24) with the equation obtained in SM:1 λ = 1 λ + 316 π ln (cid:18) Λ C µ (cid:19) , (29)or λ = λ λ π ln (cid:16) Λ C µ (cid:17) , (30)and λ = λ − λ π ln (cid:16) Λ C µ (cid:17) . (31)In the SM, the λφ model is renormalizable and produces finite scattering amplitudes and cross sections, butrenormalization theory demands that the cutoff Λ C must be taken to infinity, Λ C → ∞ [5]. Then, from (29),the renormalized coupling constant constant λ = 0. This is known as the triviality problem [6, 7, 8, 9, 10, 11].This result holds even in the limit λ → ∞ : 1 λ ∼ π ln (cid:18) Λ C µ (cid:19) . (32)In the earlier paper [4], it was demonstrated that the triviality problem for the scalar field field could beresolved in the finite QFT theory. Because Λ H = 1 / (cid:96) H is a fundamental constant to be measured, it cannotbe taken to infinity as in the case of infinite renormalization theory. Thus, we cannot take the limit (cid:96) H → δ -function limit. From Fig. 1, we observe that when we choose Λ H (cid:46) δm /m ∼ O (1) where M H = 125 GeV. From Fig. 1, weobserve that for Λ H > µ , we avoid a Landau pole and, in particular, for 700 < Λ H < µ above Λ H as a measurement probe of the running of λ is attempting to make ameasurement within the finite Gaussian distribution length size (cid:96) 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 Higgsparticle interacting with another Higgs particle. This cannot describe a realistic situation, for the Higgscoupling to other particles such as the top quark (the top quark-Higgs coupling λ t ∼ O (1)) may play animportant role. Acknowledgments
Research at the Perimeter Institute for Theoretical Physics is supported by the Government of Canadathrough industry Canada and by the Province of Ontario through the Ministry of Research and Innovation(MRI).
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