Can the Renormalization Group Improved Effective Potential be used to estimate the Higgs Mass in the Conformal Limit of the Standard Model?
F.A. Chishtie, T. Hanif, J. Jia, R.B. Mann, D.G.C. McKeon, T.N. Sherry, T.G. Steele
aa r X i v : . [ h e p - ph ] M a y Can the Renormalization Group Improved Effective Potential be used toestimate the Higgs Mass in the Conformal Limit of the Standard Model?
F.A. Chishtie , T. Hanif , J. Jia , R.B. Mann , D.G.C. McKeon , T.N. Sherry , andT.G. Steele Department of Applied Mathematics, The University of Western Ontario, London, ON N6A 5B7, Canada Department of Physics and Astronomy, The University of Western Ontario, London, ON N6A 5B7, Canada Department of Physics, University of Waterloo, Waterloo, ON N2L 3G1, Canada Department of Mathematics and Computer Science, Algoma University, Sault St. Marie, ON N6A 2G4, Canada School of Mathematics, Statistics and Applied Mathematics, NUI Galway, University Road, Galway, Ireland School of Theoretical Physics, Dublin Institute for Advanced Studies, Burlington Rd., Dublin 4, Ireland Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, SK S7N 5E2, Canada Department of Theoretical Physics, University of Dhaka, Dhaka-1000, Bangladesh
June 7, 2018
Abstract
We consider the effective potential V in the standard model with a single Higgs doublet in the limit that the onlymass scale µ present is radiatively generated. Using a technique that has been shown to determine V completelyin terms of the renormalization group (RG) functions when using the Coleman-Weinberg (CW) renormalizationscheme, we first sum leading-log (LL) contributions to V using the one loop RG functions, associated with fivecouplings (the top quark Yukawa coupling x , the quartic coupling of the Higgs field y , the SU (3) gauge coupling z , and the SU (2) × U (1) couplings r and s ). We then employ the two loop RG functions with the three couplings x , y , z to sum the next-to-leading-log (NLL) contributions to V and then the three to five loop RG functions withone coupling y to sum all the N LL . . . N LL contributions to V . In order to compute these sums, it is necessaryto convert those RG functions that have been originally computed explicitly in the minimal subtraction (MS)scheme to their form in the CW scheme. The Higgs mass can then be determined from the effective potential: the LL result is m H = 219 GeV /c decreases to m H = 188 GeV /c at N LL order and m H = 163 GeV /c at N LL order. No reasonable estimate of m H can be made at orders V NLL or V N LL since the method employed giveseither negative or imaginary values for the quartic scalar coupling. The fact that we get reasonable values for m H from the LL , N LL and N LL approximations is taken to be an indication that this mechanism for spontaneoussymmetry breaking is in fact viable, though one in which there is slow convergence towards the actual value of m H . The mass 163 GeV /c is argued to be an upper bound on m H . Keywords
Renormalization group; Effective potential; Standard model; Higgs mass; Coleman-Weinberg renormalization scheme;Radiative effects.
The leading-logarithm (LL) contribution to the effective potential V in the standard model in which there is a singlescalar field and no mass scale in the classical limit, has been used to estimate the Higgs mass to be m H = 224 GeV /c [1]. Subsequent investigations indicate that contributions beyond LL to V do not destabilize this result [2]. In thispaper we propose to significantly improve the methods used in refs. [1, 2] and compute the resulting modification tothe estimate of m H . The value of m H obtained using these improvements is much more realistic.Since these results were obtained, it has been established that when the CW renormalization scheme is used tocompute V , all N p LL contributions to V can be computed using the ( p + 1) loop RG functions when there is a singlescalar field φ without a classical mass term for this scalar in the action [3]. ∗ To whom correspondence should be addressed. Email: [email protected]
1e first show how these techniques can be used to refine the approach of [1, 2]. In doing so, we overcome severalshortcomings of the original calculation. First of all, the RG functions we use are those appropriate to the CWrenormalization scheme, not the minimal subtraction (MS) scheme. This conversion from the MS scheme (in whichthe RG functions were originally computed) to the CW scheme was not carried out in [1, 2]. Next, we show how the N p LL contributions to V can be expressed exactly in terms of the ( p + 1) loop CW RG functions. This shows thatonce the ( p + 1) loop CW RG functions are known, we have an exact expression for the ( p + 1) loop contributionsto V without having to compute any Feynman diagrams and, in addition, we can sum all the N p LL contributionsto V coming from all orders in the loop expansion. In [1, 2] these contributions were only given as a power seriesin the couplings x , z , r and s . Finally, we compute the counter-term that takes into account all log-independentcontributions to V beyond the N p LL order in a more consistent way than was done in [1, 2]; rather than fixingthis counter-term by the LL calculation and then using this value at higher order, we determine the value of thiscounter-term at each order separately thereby taking into account how the value of the coupling y is adjusted. It isthe methods of ref. [3] that allow us to fix all log independent contributions to V in terms of the RG functions whenusing the CW scheme. Our analytic approach supplements numerical techniques for investigating V using the RGequation (see e.g., ref. [45]).In the next section we review how N p LL contributions to V can be computed in terms of the RG functions whenthe CW renormalization scheme is used, first considering the case in which there is a single O ( N ) scalar field withonly a quartic self coupling and no classical mass term in the Lagrangian. The only mass scale in such a theoryis radiatively induced. This is then extended so that the scalar couples to other fields (both vectors and spinors).The details of the solution at N LL are presented in Appendix 1 along with an explanation of how the methodologycan be extended to N LL and higher-order. Appendix 2 presents a method of computing terms in the derivativeexpansion of the one loop effective action.We have employed the CW renormalization scheme, as in this scheme all logarithmic dependence on the externalfield comes through a single form of logarithm, ln (cid:0) φ /µ (cid:1) . Having this single logarithm simplifies the ansatz wemake for V when there are multiple couplings (see eq. (19) below), making it possible to find V in terms of the CWRG functions. If there are multiple couplings (say x and y ) then both ln (cid:0) xφ /µ (cid:1) and ln (cid:0) yφ /µ (cid:1) arise when usingthe MS renormalization scheme. This complicates the ansatz one has for V , making it no longer feasible to find V in terms of the MS RG functions. Furthermore, one must compute the radiative corrections dependent on φ to thekinetic term ( ∂ µ φ ) in the effective Lagrangian when determining the radiatively generated Higgs mass m H ; this isunknown (and presumably non-trivial) in the MS scheme, whereas in the CW scheme it is defined to be equal to oneat the value of φ that minimizes V (see eq. (18) below). For these reasons we use the CW scheme in our analysis.We also note that the inclusion of a quadratic mass term m φ for the O (4) scalar field in the classical actionresults in multiple forms of the logarithm occurring in the ansatz for V (see ref. [9]) and also necessitates considerationof a “cosmological term” (see ref. [44]). These factors considerably complicate employing the RG equation to find the N p LL contributions to V ; we thus restrict ourselves to the classically conformal case m = 0 as originally suggestedin ref. [4].We then discuss the conversion of the RG functions from the MS scheme, in which they have been originallycomputed, to the CW scheme, which is necessary to implement our procedure for computing the N p LL contributionto V . We finally apply these results to the simplest version of the standard model in which there is a single scalarwhich is an SU (2) doublet and which has no mass at the classical level. The resulting expression for the effectivepotential at N LL order leads to an estimate of 163 GeV /c for the mass of the Higgs Boson. We regard this asan upper limit on the Higgs mass as lower order calculations lead to estimates that are considerably higher thanthis. In any case, the proposal [4] that the Higgs mechanism is a consequence of radiative corrections to the effectivepotential in the conformally invariant classical limit of the standard model is seen to be viable.We note that the potential V being considered here is the sum of all one particle irreducible (1PI) diagramswith external scalar fields whose momentum vanishes. This 1PI potential has been argued to be distinct from the“effective potential”, a quantity shown in ref. [25] to be convex and real. The relationship between the 1PI potentialand the effective potential is discussed in refs. [26, 27] and reviewed in refs. [28, 29, 30]. However, resolution of theconvexity problem continues to be debated in the literature (see refs. [31, 32, 33]). The most recent examination ofthe convexity problem explores the distinctions between the Euclidean and Minkowskian formulations of the effectivepotential [34].Although our work adopts the conventional approach of ascribing physical meaning to the 1PI potential [26,27, 28, 29], it is important to note that our Higgs mass predictions in the standard model rely upon only the local properties of the 1PI potential near the minimum as extracted from the RG equation. Since this minimum occurs atnon-zero field values, the minimum corresponds to the qualitative non-perturbative form of a spontaneous symmetrybreaking effective potential [35] and provides the lower bound on the region where the effective potential and 1PIpotential coincide [26, 27, 29]. Therefore our analysis is not in conflict with Ref. [31], which argues that the 1PI and2ffective potential must agree near the minimum and advocates the use of RG methods.Finally we note that non-perturbative approaches are not isolated from the convexity problem. For example,the constraint effective potential [36] in lattice approaches is non-convex at finite volumes [37], and lattice resultsare found to agree with the perturbative 1PI potential in appropriate regions of parameter space [38]. Functionalflows of the exact renormalization group can be used to calculate an effective average action [39, 40] and convexityconstrains the regulators used in various truncation schemes used in these methods [41]. Other alternatives to theeffective potential include the Gaussian effective potential [43] which is well-suited to variational techniques.
We begin by considering an O ( N ) scalar field φ with a classical potential V cl V cl = λφ = π yφ (1)where λ is the usual scalar coupling constant but y is more useful as it removes explicit factors of π in RG functions.The coupling y is renormalized so that the effective potential V satisfies the CW renormalization condition [4] d V ( φ ) dφ (cid:12)(cid:12) φ = µ = 24 π y (2)is satisfied. Radiative corrections to the effective potential [4, 5, 6, 28] with this renormalization condition take theform V ( y, φ, µ ) = π ∞ X n =0 n X m =0 y n +1 T nm L m φ (3)where L = ln (cid:16) φ µ (cid:17) . In order that there be no net dependence on the renormalization scale parameter µ , V mustsatisfy µ dVdµ = 0 = (cid:18) µ ∂∂µ + β ( y ) ∂∂y − φγ ( y ) ∂∂φ (cid:19) V (4)where µ dydµ = β ( y ) = ∞ X n =2 b n y n (5)and µφ dφdµ = − γ ( y ) = − ∞ X n =1 g n y n . (6)The RG equation (4) and its solution for V in eq. (3) corresponds to the situation where there is no quadraticterm for the scalar field, consistently maintaining the massless nature of the theory. In particular, extension tomassive theories is achieved by including a mass term and anomalous mass dimension into the RG equation (4) (seeRef. [44] for an analysis of a single-component massive scalar theory). It is therefore not necessary for us to imposethe V ′′ ( φ = 0) = 0 renormalization condition used by Coleman & Weinberg [4] to eliminate quadratic divergences.Furthermore, vacuum graphs do not generate divergences that are eliminated by renormalization of the cosmologicalterm [44]. As outlined below, we also do not introduce quadratic counter-terms into the phenomenological analysisof V .If now the N p LL contribution to V in eq. (3) is defined to be V N p LL = π y p +1 S p ( yL ) φ where S n ( yL ) = ∞ X m =0 T n + m,m ( yL ) m (7)so that V = π ∞ X n =0 y n +1 S n ( yL ) φ (8)then eq. (4) is satisfied at order y n +2 provided S n ( ξ ) satisfies (cid:20) ( − b ξ ) ddξ + b − g (cid:21) S = 0 (9) The average effective action is calculated for scalar QED in Ref. [42]. (cid:20) ( − b ξ ) ddξ + ( n + 1) b − g (cid:21) S n + n − X m =0 (cid:20) − g n − m + b n − m +2 ξ ddξ + ( m + 1) b n − m +2 − g n − m +1 (cid:21) S m = 0 (10)with the boundary condition S n (0) = T n . (11)Thus V can be determined by solving the coupled equations (9, 10) provided the boundary values T n are known.These are fixed by the CW condition of eq. (2); since L = 0 when φ = µ eqs. (2, 8) together imply that24 y = ∞ X k =0 y k +1 (cid:2) y S ′′′′ k (0) + 80 y S ′′′ k (0) + 140 y S ′′ k (0) + 100 yS ′ k (0) + S k (0) (cid:3) . (12)Since g = 0, together (9) and (12) lead to T = 1 (13) S ( ξ ) = 1 w (14)where w = 1 − b ξ . Eq. (12) then gives T = − b (15)so that eq. (10) be solved when n = 1 S ( ξ ) = 4 g b w − g + b b w − b b w ln | w | = 14 w + (cid:18)
14 ln | w | − (cid:19) w (for N = 4) . (16)This process can be continued indefinitely; S p ( ξ ) can be determined in terms of b . . . b p +2 , g . . . g p +1 where theseRG function coefficients are those appropriate to the CW scheme.If in addition to y there are other couplings g i ( i = 1 . . . N ) (Yukawa, gauge etc.) in the theory then the CWrenormalization condition (2) must be supplemented by additional conditions. For example, in massless scalarelectrodynamics in which a complex scalar φ is coupled to a U (1) gauge field A µ with coupling e , then the effectiveaction takes the form [4]Γ = Z d x (cid:20) − V ( φ ) + 12 Z ( φ ) | ( ∂ µ − ieA µ ) φ | − H ( φ )( ∂ µ A ν − ∂ ν A µ ) + . . . (cid:21) . (17)Infinities arise when computing V , Z and H and so in addition to (2) one requires renormalization conditions whichwe take to be H ( φ = µ ) = 1 = Z ( φ = µ ) . (18)Application of the RG equation to determine higher order corrections to Z ( φ ) is discussed in ref. [20].Suppose that x and y are the only two couplings. (It is easy to extend our considerations to include more thantwo.) The expansion of eq. (3) now generalizes to V = π ∞ X n =1 n + k X r =0 ∞ X k =0 T n + k − r,r,k y n + k − r x r L k (19)and V satisfies the RG equation (cid:18) µ ∂∂µ + β x ∂∂x + β y ∂∂y − φγ ∂∂φ (cid:19) V = 0 . (20)The RG functions are β x = µ dxdµ = ∞ X n =2 β xn = ∞ X n =2 n X r =0 b xn − r,r x r y n − r (21) β y = µ dydµ = ∞ X n =2 β yn = ∞ X n =2 n X r =0 b yn − r,r x r y n − r (22) γ = − µφ dφdµ = ∞ X n =1 γ n = ∞ X n =1 n X r =0 g n − r,r x r y n − r . (23)4he N p LL contribution to V is now given by V N p LL = π ∞ X k =0 p kk + p +1 L k φ (24)where p kn ( x, y ) = n X r =0 T n − r,r,k y n − r x r ( n ≥ k + 1) (25)so that V = ∞ X p =0 V N p LL . (26)The CW condition of eq. (2) now shows that for all n yδ n = 24 p n + 100 p n + 280 p n + 480 p n + 384 p n . (27)Furthermore, the RG equation (20) leads to ∞ X n =1 n − X k =0 " − kp kn L k − + ∞ X m =2 (cid:18) β xm ∂∂x + β ym ∂∂y (cid:19) p kn L k − ∞ X m =1 (cid:0) γ m p kn L k + 2 kγ m p kn L k − (cid:1) = 0 . (28)Together, (27, 28) fix V in terms of the CW RG functions.We employ a novel way of treating the sums in eq. (24), which involves using the method of characteristics [3].Beginning with the definition w kn ( x ( t ) , y ( t ) , t ) = p kn ( x ( t ) , y ( t )) exp (cid:20) − Z t γ ( x ( τ ) , y ( τ )) dτ (cid:21) (29)where dx ( t ) dt = β x ( x ( t ) , y ( t )) (30) dy ( t ) dt = β y ( x ( t ) , y ( t )) (31)with x (0) = x , y (0) = y we find that ddt w kn ( x, y, t ) = (cid:18) β x ( x, y ) ∂∂x + β y ( x, y ) ∂∂y − γ ( x, y ) (cid:19) w kn ( x, y, t ) . (32)Eq. (28) is satisfied to order n − L and n + 1 in the couplings x and y provided p nn +1 = 12 n (cid:18) β x ∂∂x + β y ∂∂y − γ (cid:19) p n − n (33)so that by eqs. (29, 32, 33) w nn +1 ( x, y, t ) = 12 n ddt w n − n ( x, y, t ) . (34)If now V N p LL ( x, y, t ) = π ∞ X k =0 w kk + p +1 ( x, y, t ) L k φ (35)so that if t = 0 V N p LL ( x, y,
0) = V N p LL (36)then by (29) V LL ( x ( t ) , y ( t ) , t ) = π ∞ X n =0 L n n n ! d n dt n w ( x ( t ) , y ( t ) , t ) φ = π w ( x ( t + L , y ( t + L , L φ (37)5nd hence by (36) we finally have a closed form expression for V LL . V LL = π w ( x ( L , y ( L , L φ . (38)The detailed computation of V NLL presented in Appendix 1 gives eq. (142) V NLL = π φ exp " − Z L/ dτ γ ( x i ( τ )) p (cid:18) x i (cid:18) L (cid:19)(cid:19) + Z L/ dτ h(cid:16) − γ ( x i ( τ )) β x i ( x i ( τ )) + β x i ( x i ( τ )) (cid:17) U ij (0 , τ ) i . " U jk (cid:18) L , (cid:19) ∂∂x k ( L ) p (cid:18) x i (cid:18) L (cid:19)(cid:19) +4 Z L/ dτ (cid:2) γ ( x i ( τ )) − γ ( x i ( τ )) (cid:3) p (cid:18) x i (cid:18) L (cid:19)(cid:19)) , (39)where by eqs. (27, 33, 116) p = y , p = 12 β y − γ y , p = − p , (40)and by eqs. (129–133) ddt U ( t,
0) = U ( t, M (41) U − ( t,
0) = U (0 , t ) = 1 + ∞ X n =1 ( − n Z t dτ . . . Z τ n − dτ n [ M ( τ ) . . . M ( τ n ) ] (42)and M ij = ∂β x j ∂ ¯ x i . (43)The techniques used to find V NLL in eq. (39) can be extended to obtain V N LL . However, since the three loopRG functions needed for this extension have not been computed for the standard model, we will not pursue thiscalculation further.We now will discuss how the CW RG functions can be found if the MS RG functions are known. The RG functions have been computed using dimensional regularization and minimal subtraction to five loop orderin an O ( N ) scalar theory [7] and to two loop order in the standard model [8]. We will now examine how from theseknown results one can find the RG functions in the CW renormalization scheme.First, we quote the MS values of the O ( N ) scalar model of eq. (1) to five loop order [7]˜ β ( y ) = N + 82 y −
34 (3 N + 14) y + 164 (cid:2) N + 922 N + 2960 + 96(5 N + 22) ζ (3) (cid:3) y − (cid:18) (cid:19) y (cid:20) − N + 6320 N + 80456 N + 196648 + 96 (cid:0) N + 764 N + 2332 (cid:1) ζ (3) − N + 22)( N + 8) ζ (4) + 1920 (cid:0) N + 55 N + 186 (cid:1) ζ (5)] + 43 (cid:18) (cid:19) y " N + 12578 N + 808496 N + 6646336 N + 13177344+ 16 (cid:0) − N + 1248 N + 67640 N + 552280 N + 1314336 (cid:1) ζ (3) + 768 (cid:0) − N − N + 446 N + 3264 (cid:1) ζ (3) − (cid:0) N + 1388 N + 9532 N + 21120 (cid:1) ζ (4) + 256 (cid:0) N + 7466 N + 66986 N + 165084 (cid:1) ζ (5) − N + 8) (cid:0) N + 55 N + 186 (cid:1) ζ (6) + 112896 (cid:0) N + 189 N + 526 (cid:1) ζ (7) + O ( y ) (44)6nd ˜ γ ( y ) = N + 216 y − ( N + 2)( N + 8)128 y + (cid:18) (cid:19) y N + 2) (cid:2) (cid:0) − N + 18 N + 100 (cid:1)(cid:3) − (cid:18) (cid:19) y N + 2) " N + 296 N + 22752 N + 77056 − N − N + 64 N + 184) ζ (3)+1152(5 N + 22) ζ (4) + O ( y ) . (45)We next provide the two loop RG functions in the standard model in which there is a single scalar doublet withno mass term for this field in the classical action. The quartic scalar coupling y appears in eq. (1); the other couplingsare the top quark Yukawa coupling x = g t π (46)the SU (3) coupling z = g π (47)and the SU (2) × U (1) couplings r = g π (48) s = g π . (49)To two loop order the RG functions in this simplest version of the standard model [8] in the MS renormalizationscheme are ˜ β x =˜ µ dxd ˜ µ = (cid:20) x − xz − xr − xs (cid:21) + " − x + 131128 x s + 225128 x r + 92 x z − x y + 11871728 xs − xrs + 1972 xsz − xr + 98 xrz − xz + 34 xy + . . . (50)˜ β y =˜ µ dyd ˜ µ = (cid:20) y + 3 xy − x − yr − ys + 332 s + 316 rs + 932 r (cid:21) + " − y − xy + 274 y r + 94 y s − x y + 5 xyz + 4532 xyr + 8596 xys − yr + 3964 yrs + 629384 ys + 158 x − x z − x s − xr + 2132 xrs − xs + 305256 r − r s − rs − s + . . . (51)˜ β z = ˜ µ dzd ˜ µ = (cid:20) − z (cid:21) + (cid:20) sz + 916 rz − z − xz (cid:21) + . . . (52)˜ β r = ˜ µ drd ˜ µ = (cid:20) − r (cid:21) + (cid:20) r s + 3548 r + 32 r z − xr (cid:21) + . . . (53)˜ β s = ˜ µ dsd ˜ µ = (cid:20) s (cid:21) + (cid:20) s + 916 rs + 116 zs − xs (cid:21) + . . . (54)and˜ γ = − ˜ µφ dφd ˜ µ = (cid:20) x − r − s (cid:21) + (cid:20) y − x + 54 xz + 45128 xr + 85384 xs − r + 9256 rs + 411536 s (cid:21) + . . . (55)In the case of there being only an O ( N ) scalar field φ , we follow the procedure outlined in refs. [3, 21] to convertfrom the RG functions of eqs. (44, 45) to those appropriate to the CW scheme. In the MS scheme, the computationresults in an expansion of V that is similar to that of eq. (3), V = π ∞ X n =0 n X m =0 y n +1 ˜ T nm ˜ L m φ (56)7here now ˜ L = ln (cid:16) yφ ˜ µ (cid:17) . If the RG scale ˜ µ in the MS scheme is rescaled˜ µ = y / µ (57)where µ is the RG scale in the CW scheme, then the form of the expansion of eq. (56) becomes that of eq. (3). Finiterenormalizations of the form y → y (1 + a y + a y + . . . ) (58) φ → φ (1 + b y + b y + . . . ) (59)may then be required to adjust the coefficients ˜ T n in eq. (56) so that the CW RG condition of eq. (2) is satisfied, butthis can be done without altering ˜ T nm ( m >
0) and hence the terms in V that fix the RG functions are not changed[10].With the rescaling of eq. (57) β ( y ) = µ ∂y∂µ = (˜ µy − / ) (cid:18) ∂ ( y / µ ) ∂µ (cid:19) ∂y∂ ˜ µ = ˜ β ( y ) / (cid:16) − ˜ β ( y ) / (2 y ) (cid:17) (60)and similarly γ ( y ) = ˜ γ ( y ) / (cid:16) − ˜ β ( y ) / (2 y ) (cid:17) . (61)Eqs. (60, 61) allow one to pass from the MS RG functions of eqs. (44, 45) to the CW RG functions.It is somewhat more complicated to convert the RG functions of eqs. (45–50) to the CW scheme since more thanone type of logarithm arises when V is computed using the MS renormalization scheme. A computation of V in theCW scheme would allow one to infer the CW RG functions, but to obtain in this way the RG functions to order n , one must compute V to order ( n + 1) [10]. Since V in the standard model has only been computed to secondorder [11] one cannot determine the CW RG functions to two loop order from V directly; other contributions to theeffective action must be considered.Suppose the couplings in a theory are g i (with g i = ( x, y, z, r, s ) in the standard model) and that there is onescalar field φ . When computing V using MS, logarithms of the form ˜ L i = ln (cid:0) g i φ / ˜ µ (cid:1) arise. At one loop order inMS, only these types of logarithms occur; beyond one loop order other more complicated logarithms arise [11] butdo not affect our discussion of how the MS and CW RG functions are related at two loop order. As in refs. [3, 9] weassociate a separate renormalization scale κ i with each of these logarithms so that now˜ L i = ln (cid:18) g i φ κ i (cid:19) . (62)A rescaling similar to that of eq. (57) κ i = g / i µ (63)leads to β g i = µ ∂g i ∂µ = X j ˜ β g i j (cid:18) β g j g j (cid:19) (64) γ = − φµ ∂φ∂µ = X j ˜ γ j (cid:18) β g j g j (cid:19) (65)where ˜ β g i j = κ j ∂g i ∂κ j (66)˜ γ j = − κ j φ ∂φ∂κ j . (67)Again, µ is the CW mass parameter. We also see that˜ β g i = X j ˜ β g i j (68)˜ γ = X j ˜ γ j (69)8here ˜ β g i and ˜ γ are the MS RG functions.We now will use eqs. (64, 65) to find the CW RG functions to two loop order in the standard model, restrictingourselves to the limiting case in which only the three dominant couplings g = x , g = y and g = z are considered.If we use Roman numeral subscripts with the RG functions to denote the number of coupling constants present in aperturbative expansion (e.g., ˜ β x II is the term in the expansion of the β function for x in the MS scheme associatedwith the mass scale κ that has two powers of the coupling), then by eqs. (64–67) we see that β g i II = ˜ β g i II (70) γ I = ˜ γ I ; (71)that is at lowest order the RG functions in the CW and MS schemes are the same. It also follows that β g i III = ˜ β g i III + X j ˜ β g i j II β g j II g j (72) γ II = ˜ γ II + X j ˜ γ j I β g j II g j . (73)Eqs. (72, 73) show that apart from standard RG functions, only the one loop multi-scale RG quantities ˜ β g i j II and˜ γ j I are needed to obtain the two loop CW RG functions β g i III and γ II .To find ˜ γ j I we note that the one loop scalar self energy in the standard model (with no classical mass term forthe scalar and just the couplings x , y and z ) only has a contribution coming from the top quark loop. Consequentlythe term Z ( φ )( ∂ µ φ ) in the effective action only receives a logarithmic contribution of the form ln (cid:0) xφ / ˜ µ (cid:1) and sowe see that ˜ γ I = ˜ γ I (74)˜ γ I = ˜ γ I = 0 . (75)To obtain ˜ β y I , ˜ β y II and ˜ β y II , we note that at leading log one loop order in the model we are considering [8], V is given in the MS scheme by V = π (cid:20) y + (cid:18) y ln yφ ˜ µ − x ln xφ ˜ µ (cid:19)(cid:21) φ . (76)If the RG equation of eq. (20) is to be satisfied for each of the three mass scales κ j introduced in eq. (62), we findthat consistency with eqs. (74, 75) occurs if ˜ β y II = − x + 3 xy (77)˜ β y II = 6 y (78)and ˜ β y II = 0 . (79)Determining ˜ β z II , ˜ β z II and ˜ β z II is most easily done by considering the one loop contribution to the term − H ( φ ) F in the effective action where F aµν is the SU (3) field strength. As only a quark loop can contribute atone loop order to H ( φ ), then the only logarithmic contribution to H ( φ ) at one loop order is ln (cid:0) xφ / ˜ µ (cid:1) in the MSscheme. However, H ( φ ) dictates the function ˜ β z on account of gauge invariance [12] and so˜ β z II = ˜ β zII (80)and ˜ β z II = ˜ β z II = 0 . (81)For ˜ β x II , ˜ β x II and ˜ β x II we note that the scalar-quark-quark vertex only receives a logarithmic contribution atone loop order of the form ln (cid:0) xφ / ˜ µ (cid:1) and hence ˜ β x II = ˜ β xII (82)˜ β x II = ˜ β x II = 0 . (83)9ogether, eqs. (74–83) result in eqs. (72, 73) yielding to two loop order in the CW scheme β x = (cid:20) x − xz (cid:21) + (cid:20) − x + 92 x z − x y − xz + 34 xy (cid:21) + 12 x (cid:20) x − xz (cid:21) + . . . = (cid:20) x − xz (cid:21) + (cid:20) x − x z − x y + 34 xy − xz (cid:21) + . . . (84) β y = (cid:20) y + 3 xy − x (cid:21) + (cid:20) − y − xy − x y + 5 xyz + 158 x − x z (cid:21) + 12 x (cid:20) − x + 3 xy (cid:21) (cid:20) x − xz (cid:21) + 12 y (cid:2) y (cid:3) (cid:20) y + 3 xy − x (cid:21) + . . . = (cid:20) y + 3 xy − x (cid:21) + (cid:20) − y + 316 x + x z − xyz − x y (cid:21) + . . . (85)(which is the same result as is obtained from eq. (60) if x = z = 0) β z = (cid:20) − z (cid:21) + (cid:20) − z − xz (cid:21) + 12 x (cid:20) − z (cid:21) (cid:20) x − xz (cid:21) + . . . = (cid:20) − z (cid:21) + (cid:20) z − xz (cid:21) + . . . (86)and γ = (cid:20) x (cid:21) + (cid:20) − x + 38 y + 54 xz (cid:21) + 12 x (cid:20) x (cid:21) (cid:20) x − xz (cid:21) + . . . = (cid:20) x (cid:21) + (cid:20) x + 38 y − xz (cid:21) + . . . (87)(Exact solutions for the one loop characteristic functions x ( t ), y ( t ), z ( t ) appear in [13].)With these CW RG functions we can compute V NLL using eq. (39) in the model we are considering.
We now show how the results of the previous two sections can be applied to the standard model in order to estimatethe mass of the Higgs Boson. We only consider the case in which there is a single Higgs doublet with no classicalmass term.As was pointed out in [1, 2], there are three things to consider. First of all, we have the CW renormalizationconditions of eqs. (2, 18). Next there is the stability condition ddφ V ( φ = µ ) = 0 . (88)This means that we identify µ with the vacuum expectation value of φ , that is µ = 2 − / G − / F . Once these tworequirements are satisfied, we can compute the Higgs mass by the formula m H = d V ( φ = µ ) dφ /Z ( φ = µ ) . (89)With the renormalization condition of eq. (18) this just reduces to m H = d V ( φ = µ ) dφ . (90)If V is expanded in the form V = ∞ X p =0 V N p LL (91)10here V N p LL is the N p LL contribution to V , then we begin by estimating V by V m = m X p =0 V N p LL + π K m φ . (92)The term π K m φ in eq. (92) represents the parts of V coming from those terms in eq. (91) beyond N m LL whichcan be determined by imposing eq. (2) — the renormalization condition. As is discussed in section two above, V N p LL can be determined in terms of the CW RG functions if they are known to p + 1 loop order. From section two then, V LL can be found using all five couplings ( x, y, z, r, s ), V NLL can be found using the three couplings ( x, y, z ) andfinally V N LL , V N LL and V N LL can be found using the single coupling y .The role of K m in eq. (92) is to ensure that the CW renormalization condition of eq. (2) is satisfied. It is a“counter-term”; more explicitly in terms of the quantities p kn introduced in eq. (25) (or the generalization of thisexpression to accommodate more than two couplings) K m = ∞ X n = m +2 p n . (93)Eqs. (12) and (27) on their own only ensure that eq. (2) is satisfied up to a finite order m in the coupling constantexpansion; the inclusion of the counter-term ensures that eq. (2) is satisfied to all orders. Once expressions for V LL . . . V N m LL have been given in terms of the appropriate CW RG functions, there are still two unknowns: thecounter-term K m and the quartic scalar coupling y . These two are fixed by conditions (2) and (88), then V m is usedin conjunction with eq. (90) to estimate m H .More explicitly, V LL is given by eq. (38) with eq. (29) leading to V LL = π p (cid:18) x (cid:18) L (cid:19) , y (cid:18) L (cid:19) , z (cid:18) L (cid:19) , r (cid:18) L (cid:19) , s (cid:18) L (cid:19)(cid:19) exp " − Z L/ dτ γ ( x ( τ ) , . . . , s ( τ )) φ . (94)We see by eq. (27), p = y and by eqs. (55, 71), γ = x − r − s .When one computes derivatives of the characteristic functions x ( t ) . . . s ( t ) when evaluating V ′ LL , V ′′ LL and V ′′′′ LL asrequired by eqs. (2, 88, 89), the one loop contributions to ˜ β x . . . ˜ β s in eqs. (50 – 54) are to be used as at one looporder the CW and MS RG functions are the same.For V NLL we need RG functions in the CW renormalization scheme to two loop order. These are given by eqs. (84– 86) for the limiting case in which the standard model with only the three couplings ( x, y, z ) is being considered.These are used in conjunction with V NLL in eq. (39). In this equation, we have w (cid:18) x (cid:18) L (cid:19) , y (cid:18) L (cid:19) , z (cid:18) L (cid:19)(cid:19) = y (cid:18) L (cid:19) exp " − Z L/ dτ (cid:18) x ( τ ) (cid:19) (95)and since by eqs. (40, 84–87) p = y p = 3 y − x p = − p (96)we also have w = (cid:20) − y (cid:18) L (cid:19) + 258 x (cid:18) L (cid:19)(cid:21) exp " − Z L/ dτ (cid:18) x ( τ ) (cid:19) . (97)For consistency, the derivatives of x ( t ) , y ( t ) , z ( t ) that arise when computing V ′ NLL , V ′′ NLL and V ′′′′ NLL are given by theone loop contributions to β x , β y , β z occurring in eqs. (84 – 86).Finally, for V N LL , V N LL and V N LL we have at our disposal only the CW RG functions associated with thesingle scalar coupling y . These RG functions are found by combining eqs. (44, 45, 60, 61). Using them, the functions11 . . . S appearing in eq. (8) are given by S ( ξ ) = 14 w + (cid:18) − | w | − ζ (3) (cid:19) w + (cid:18)
116 ln | w | − | w | + 359116 + 212 ζ (3) (cid:19) w (98) S ( ξ ) = (cid:18) − ζ (3) − (cid:19) w + (cid:18) − π
40 + 116 ln | w | + 3654 ζ (5) + 120564 + 2398 ζ (3) (cid:19) w + (cid:18) | w | − ζ (3) ln | w | + 273 ζ (3) − | w | (cid:19) w + (cid:18) π
40 + 164 ln | w | − − ζ (3) + 271916 ln | w | − | w | + 638 ζ (3) ln | w | − ζ (5) (cid:19) w (99)and S ( ξ ) = (cid:18) − π
160 + 458 ζ (3) − ζ (5) (cid:19) w + " π − ln | w | − ζ (5) − ζ (3) + 1398 ζ (3) − ζ (7) − ζ (3) ln | w | − π w + " − ζ (3) − ζ (5) − | w | ζ (5) + 4414 ζ (3) + 1203128 ln | w | + 164 ln | w | + 23916 ln | w | ζ (3) + 721 π − π
80 ln | w | w + " − | w | ζ (3) − | w | + 1256 ln | w | − ζ (3)+ 329716 ln | w | ζ (3) − | w | − ζ (5) − ζ (3) + 7 π w + " − π − π
320 + 7 π
40 ln | w | + 51712991384 + 16258 ζ (3) + 1256 ln | w | − | w | ζ (3)+ 3689732 ζ (7) + 150592196 ζ (3) − | w | + 50084996 ζ (5) − | w | + 6316 ln | w | ζ (3) + 43815512 ln | w | − | w | ζ (5) w . (100)With one coupling, we have V N p LL = π y p +1 S p ( yL ) φ for p = 2 , , x, z, r and s at the mass scale v . The couplings x, z, r and s are defined in terms of the Yukawa and gauge couplings g t , g , g and g by eqs. (46 – 49). These in turn are related to the measured quantities m t (the top quark mass), θ w (the weak angle), M W (the W - Boson mass), α s (the strong structure constant) and α (the fine structure constant),all of which are known at the mass scale set by the Z -Boson. These relations are x = α π (cid:18) m t M W sin θ w (cid:19) (101) z = α s π (102) r = απ sin θ w (103) s = απ cos θ w . (104)where the subscript 0 means that these are evaluated at the mass of the Z -Boson. From the Particle Data Group [14],at the mass of the Z -Boson (91.1876 GeV /c ), α = 1 / . α s = . θ w = . M w = 80 . GeV /c and12 t = 171 . GeV /c . It is now necessary to evaluate these couplings at the vacuum expectation value v = 2 − / G − / F (taking G F to be 1 . × − (cid:0) GeV / c (cid:1) − ). To do this, we use the one loop limit of the RG equations that followfrom eqs. (50 – 54) as a suitable approximation µ dxdµ = 94 x − xz (105) µ dzdµ = − z (106) µ drdµ = − r (107) µ dsdµ = 4112 s . (108)Eqs. (106 – 108) have solutions [13] z = z z ln (cid:16) µµ (cid:17) (109) r = r r ln (cid:16) µµ (cid:17) (110) s = s − s ln (cid:16) µµ (cid:17) . (111)Dividing eq. (105) by eq. (106) leads to the homogeneous equation dxdz = − (cid:16) xz (cid:17) + 84 (cid:16) xz (cid:17) (112)whose solution is x = (2 / z − [(1 − / z /x )]( z/z ) − / (113)Using ( x , z , r , s , ) given by eqs. (101 – 104) at the mass scale µ = 91 . GeV /c then eqs. (109 – 111, 113)yield ( x, z, r, s ) at the mass scale µ = v = 2 − / G − / F .We can now proceed to compute the Higgs mass at each order of the expansion of V in the N p LL expanion. Witheq. (92) for V m , we use eq. (2) to fix K m in terms of y and then use eq. (88) to solve for y itself. In this paper theonly acceptable values for y are positive in order to ensure physical stability of the theory for reasonable values of φ , as will be discussed below. With these values of y (and K m ) eq. (92) can be used give an explicit expression for V m . Eq. (90) can then be used to evaluate m H . Only real and positive values of m H are acceptable. We note thatit is not necessary to find explicit results for the integrals and running couplings appearing in eqs. (94, 95, 97). Thederivatives of these expressions at φ = v that are needed to evaluate the Higgs mass are determined completely interms of the RG functions and boundary values at φ = v . Thus our methodology can be applied to very complicatedmodels and is an important tool in its own right.We present, in Table 1, the values of K m , λ = π y , m H for m = 0 , , , , x, y, z, r, s ) contribute at LL order, ( x, y, z ) contribute at N LL order and only y contributes beyond that. (The units for m H are GeV /c .) It isimportant to emphasize that the values for K m listed in Table 1 arise because of the functional dependence of K m on the coupling y ; first K m is expressed in terms of y by using eq. (2) and then y is fixed by eq. (88). m K m λ m H m = 1 or m = 3 as the values of y that follow from V and V are negative and unacceptable.This appears to be due to the large negative contribution to S and S coming from terms of order w and w respectively. 13he second derivative of the order m estimate for the effective potential, normalized to the scale v , M m = 1 v d dφ V m (cid:12)(cid:12)(cid:12)(cid:12) φ = v (114)can be viewed as a function of the scalar field coupling λ once the counter-term K m has been expressed in terms of λ . In Figure 1 we present curves for the dimensionless quantity M m ( m = 0 , ,
4) for positive values of λ while M m ispositive. The crosses on the curves correspond to the values of λ and m H found by our approach and listed in Table1 for m = (0 , , λ and m H to decrease with increasing order m .We can gain further insight on this trend in the O (4) scalar theory by extracting the counter-term from the secondderivative, normalized to the scale v , ˜ M n = 1 v d (cid:0) V n − π K n φ (cid:1) dφ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ = v . (115)For the pure scalar field theory case the resulting dimensionless expressions are shown as a function of λ in Figure2. One can see the distinction between even and odd orders in the Figure, and one can also see evidence of slowconvergence towards a result which would lie between the even and odd envelopes of the curves. Because ˜ M n represents the field-theoretical (i.e., counter-term-independent) contributions to the Higgs mass, it is evident thateven orders provide an upper bound on m H and odd orders provide a lower bound on m H . Although the lowerbound is trivial (i.e., m H = 0), this does not obviate the interpretation of m H at odd orders as an upper bound.Figure 1: The dimensionless ratio M m = v d dφ V m (cid:12)(cid:12)(cid:12) φ = v plotted as a function of λ . In this paper we have presented a systematic way of using the RG equation to sum all of the logarithms contributingto V at order N p LL in terms of the ( p + 1) order RG functions, provided we use the CW renormalization scheme andhave only one form of logarithm (here L = log (cid:2) φ /µ (cid:3) ) contributing to V . We have applied our method of analysisto the conformal limit of the standard model with a single scalar field, as was originally envisaged by Coleman andWeinberg [4]. This has led to a surprisingly interesting sequence of estimates for the Higgs mass and the quarticscalar couplings.It was not anticipated that the improvements to the approach, originally used in [1, 2], introduced in this paperand [3] would lead to a sequence of decreasing estimates for the Higgs mass as listed in Table 1 above. The valuesof these estimates suggest that increasing the order m to 6 and beyond (if that were feasible) would lead to Higgsmass estimates closer to the generally expected range of possible values. A compilation of predictions of the Higgsmass in different scenarios is given in ref. [18], and a discussion on its limits is given in ref. [15]. In our approachwe have made use of all known RG functions relevant to any part of the standard model. To make further progressusing this approach will require knowledge of RG functions at a higher loop order than is currently available.Even though we have not come up with a definitive prediction of the Higgs mass within the standard model, wefeel that our results establish the viability of the Coleman-Weinberg mechanism to generate spontaneous symmetrybreaking and to provide a mass for the Higgs scalar particle. We have done this by the use of the RG-improved14igure 2: The dimensionless quantity ˜ M n = v d ( V n − π K n φ ) dφ (cid:12)(cid:12)(cid:12)(cid:12) φ = v is plotted as a function of λ for the O (4) scalartheory. The upper curves represent the even orders ( n = 0 , ,
4) and the lower curves represent the odd orders( n = 1 , O (4) scalar field theory obtained from the standard model by setting all couplings except λ = π y to zero. We present in Table 2 the results for K m , λ and m H in this simplified model using exactly the same stepsas were used to derive the results in Table 1 for the standard model. m K m λ m H O (4) scalar theory to three significant digits.The similarity between the results of Tables 1 and 2 indicates that y is the dominant coupling in these consider-ations, much more than x , z , r or s . We note the vanishing values for K m , λ and m H in Table 2 for m = 1 ,
3. Forthis simplified model our method yields the acceptable but trivial solution λ = 0 for all values of m . In Table 2 weonly include the non-trivial solutions for m = 0 , ,
4. For these non-trivial solutions we can plot V m as a function of φ for values of φ near the VeV scale v , something which cannot be easily done in the standard model. This plot isprovided in Figure 3.Remarkably, the plots of V , V and V have the well known shape of a spontaneous symmetry breaking potentialwhen restricted to φ values near the location of the minimum. These potentials also have a singularity at φ = ± v exp (cid:0) π / λ (cid:1) ( i.e. when w = 0 ). This is significantly far from the region near the minimum.In addition to the positive and zero λ -solutions in the pure O (4) scalar field model referred to above, there arenegative λ -solutions. We have heretofore rejected negative λ -solutions as unacceptable. In contrast to the standardmodel, in the O (4) model we can plot V m as a function of φ with these negative values of λ . We show the shape of V m ( φ ) for the appropriate negative λ -values for m = 0 , , m = 1 , m cases ( m = 0 , ,
4) we note the existence of a tightly bound minimum at φ = 0, singularities at | φ | < v (since λ <
0) and local minima at φ = ± v . On the other hand, for the odd m cases ( m = 1 ,
3) we note theexistence of a highly unstable maximum at φ = 0, singularities at | φ | < v (since λ <
0) and local minima at φ = ± v .The occurrence of a singularity at w = 0 in V m may be considered pathological but away from the singular pointsthe form of V m is interesting. Whether this feature has a role to play in the standard model is an open question15igure 3: V m is plotted as a function of φ/v with λ as in Table 2..Figure 4: Shape of V m for λ < φ for m = 0 , ,
4. . Figure 5: Shape of V m for λ < φ for m = 1 ,
3. .which may be worth pursuing. It has been shown [3] in the scalar model that summing portions of the contributionsto V m beyond order m = 4 may shift such singularities.We have attempted setting K m = 0 in eq. (92), and then determining the single remaining unknown y by usingeither eq. (2) or eq. (88). Neither of these attempts leads to acceptable values of y or m H ; one must employ thecounter-term K m in eq. (92) to get reasonable values for these parameters at any value of m . In fact, by havingintroduced the counter-term, we are availing ourselves of information about terms, independent of L = log φ µ , beyondthe N p LL contribution to V . We have been unable to establish any other viable alternative to the counter-termapproach.Whereas in this paper we have used the CW renormalization scheme, preliminary investigations indicate that itmay be possible to adapt our approach to incorporate the MS renormalization scheme, at least in the single coupling O (4) scalar model. Using the MS renormalization scheme to compute the LL and N LL contributions to V whenthere is only the coupling y , realistic values of m H and y follow from eqs. (88) and (90) only if the counter-term K m of eq. (92) is included and the condition of eq. (2) is applied. Strictly speaking, eq. (2) is not part of the MSrenormalization scheme, though it might possibly be used to fix the physical value of y in the MS scheme in a wayanalogous to using the gap equation to fix a physical mass.We hope to develop this formalism in several other ways. First, inclusion of a mass term − m φ into the classicalaction should be considered [22]. Next, the inclusion of more scalars beyond an SU (2) doublet should be dealt with,as additional scalars are necessary [19] in any supersymmetric extension of the standard model. A further problemto be addressed concerns working with summing logarithmic contributions to V in the standard model using MS RGfunctions rather than converting them to the CW scheme, even though this would entail having a separate logarithmfor each coupling (see eq. (62)) and not being able to fix the terms p p +1 in eq. (24) by using some analogue of eq. (27).We would also like to see if the RG methods that have been developed could be employed in the consideration ofother physical processes [23], or the contributions to the effective action arising due to an external magnetic field[24]. 16 cknowledgements This work was largely inspired by the late Victor Elias. Roger Macleod had a useful suggestion. There was helpfulcorrespondence with C. Ford and S. Martin. NSERC (Natural Science & Engineering Research Council of Canada)provided funding for RBM and TGS. 17 eferences [1] V. Elias, R.B. Mann, D.G.C. McKeon and T.G. Steele,
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N LL and N LL Order
The computation of V NLL begins by noting that by eq. (28) p nn +2 + γ p nn +1 = 12 n (cid:20)(cid:18) β x ∂∂x + β y ∂∂y − γ (cid:19) p n − n +1 + (cid:18) β x ∂∂x + β y ∂∂y − γ (cid:19) p n − n (cid:21) . (116)so that together eqs. (29, 32, 34, 116) imply that w nn +2 = 12 n (cid:20) ddt w n − n +1 + D ( t ) w n − n (cid:21) (117)where D ( t ) = − γ (cid:18) β x ∂∂x + β y ∂∂y − γ (cid:19) + (cid:18) β x ∂∂x + β y ∂∂y − γ (cid:19) . (118)Iterating eq. (117) shows that w nn +2 = 12 n n ! (cid:20) d n dt n w + (cid:18) d n − dt n − D ( t ) + d n − dt n − D ( t ) ddt + ... + D ( t ) d n − dt n − (cid:19) w (cid:21) . (119)One can inductively prove the identity (cid:18) d n − dt n − f + d n − dt n − f ddt + . . . + ddt f d n − dt n − + f d n − dt n − (cid:19) g = d n dt n ( φg ) − φ d n dt n g ( dφdt ≡ f ) . (120)To employ eq. (120) to simplify eq. (119) we need to commute the functional derivatives appearing in D ( t ) (seeeq. (118)) through ddt so that they act on g before ddt does. (This step was not considered properly in eq. (B22) ofref. [3].) In order to do this, we first write D ( t ) in eq. (118) in the form D ( t ) = A i ∂∂x i ( t ) + B (121)where x ( t ) ≡ x ( t ) , (122) x ( t ) ≡ y ( t ) , (123) A ( x i ( t )) ≡ − γ β x + β x , (124) A ( x i ( t )) ≡ − γ β y + β y , (125)and B ( x i ( t )) ≡ γ − γ ) . (126)Furthermore, using eqs. (30, 31), ddt = β x ( x ( t ) , y ( t )) ∂∂x ( t ) + β y ( x ( t ) , y ( t )) ∂∂y ( t ) + ∂∂t ≡ Λ i ∂∂x i + ∂∂t . (127)We now note that A i ∂∂x i dfdt = A i ∂∂x i (cid:18) Λ j ∂∂x j + ∂∂t (cid:19) f = A i (cid:20)(cid:18) Λ j ∂∂x j + ∂∂t (cid:19) ∂f∂x i + ∂ Λ j ∂x i ∂f∂x j (cid:21) = A i (cid:20) ddt δ ij + ( M ) ij (cid:21) ∂f∂x j (128)where ( M ) ij = ∂ Λ j ∂x i , (129)and so by iterating we obtain A i ∂∂x i (cid:18) ddt (cid:19) p f = A i (cid:20)(cid:18) ddt + M (cid:19) p (cid:21) ij ∂f∂x j . (130)20f we now define( U ( t, ij = δ ij + ∞ X n =1 Z t dτ Z τ dτ . . . Z τ n − dτ n [ M ( τ n ) M ( τ n − ) ... M ( τ ) M ( τ ) ] ij . (131)then it is evident that ddt ( U ( t, f ) = U ( t, (cid:18) ddt + M (cid:19) f (132)and that U − ( t,
0) = U (0 , t ) = 1 + ∞ X n =1 ( − n Z t dτ . . . Z τ n − dτ n [ M ( τ ) . . . M ( τ n ) ] (133)(An operator analogous to U arises in standard perturbation theory.) Together, eqs. (129–133) show that A i ∂∂x i (cid:18) ddt (cid:19) p f = A i (cid:20) U (0 , t ) (cid:18) ddt (cid:19) p U ( t, (cid:21) ij ∂∂x j f. (134)We now find that by eqs. (120, 121, 134) (cid:18) d n − dt n − D ( t ) + d n − dt n − D ( t ) ddt + . . . + ddt D ( t ) d n − dt n − + D ( t ) d n − dt n − (cid:19) w ( x i ( t ) , t )= d n dt n (cid:16) ˜ Z j ( t ) ζ j ( x i ( t ) , t ) (cid:17) − ˜ Z j ( t ) d n dt n ζ j ( x i ( z ) , t ) + d n dt n (cid:16) ˜ B ( t ) w ( x i ( t ) , t ) (cid:17) − ˜ B ( t ) d n dt n w ( x i ( t ) , t ) , (135)where ˜ Z j ( t ) ≡ (cid:18)Z t dτ A i ( x i ( τ )) U ij (0 , τ ) (cid:19) (136)˜ ζ j ( x i ( t ) , t ) ≡ U jk ( t, ∂∂x k ( t ) w ( x i ( t ) , t ) (137)and ˜ B ( t ) = Z t dτ B ( x i ( τ )) . (138)Upon combining eqs. (35, 119, 135) we obtain V NLL ( x i ( t ) , t ) = π φ ∞ X k =0 k ! (cid:18) L (cid:19) k "(cid:18) ddt (cid:19) k w ( x i ( t ) , t ) + (cid:18) ddt (cid:19) k (cid:16) ˜ Z j ( t ) ζ j ( x i ( t ) , t ) (cid:17) − ˜ Z j ( t ) (cid:18) ddt (cid:19) k ζ j ( x i ( t ) , t ) + (cid:18) ddt (cid:19) k (cid:16) ˜ B ( t ) w ( x i ( t ) , t ) (cid:17) − ˜ B ( t ) (cid:18) ddt (cid:19) k w ( x i ( t ) , t ) . (139)If we now employ Taylor’s theorem with eq. (139), it follows that V NLL = π φ (cid:20) w (cid:18) x i (cid:18) t + L (cid:19) , t + L (cid:19) + (cid:18) ˜ Z j (cid:18) t + L (cid:19) − ˜ Z j ( t ) (cid:19) ζ j (cid:18) x i (cid:18) t + L (cid:19) , t + L (cid:19) + (cid:18) ˜ B (cid:18) t + L (cid:19) − ˜ B ( t ) (cid:19) w (cid:18) x i (cid:18) t + L (cid:19) , t + L (cid:19)(cid:21) (140)and so by eq. (36) V NLL = π φ (cid:20) w (cid:18) x i (cid:18) L (cid:19) , L (cid:19) + ˜ Z j (cid:18) L (cid:19) ζ j (cid:18) x i (cid:18) L (cid:19) , L (cid:19) + ˜ B (cid:18) L (cid:19) w (cid:18) x i (cid:18) L (cid:19) , L (cid:19)(cid:21) (141)or, more explicitly V NLL = π φ exp " − Z L/ dτ γ ( x i ( τ )) p (cid:18) x i (cid:18) L (cid:19)(cid:19) + Z L/ dτ h(cid:16) − γ ( x i ( τ )) β x i ( x i ( τ )) + β x i ( x i ( τ )) (cid:17) U ij (0 , τ ) i . " U jk (cid:18) L , (cid:19) ∂∂x k ( L ) p (cid:18) x i (cid:18) L (cid:19)(cid:19) +4 Z L/ dτ (cid:2) γ ( x i ( τ )) − γ ( x i ( τ )) (cid:3) p (cid:18) x i (cid:18) L (cid:19)(cid:19)) . (142)21e have used the fact that ˜ B (0) = 0 = ˜ Z i (0). V N LL can be computed using the approach used to obtain V NLL . Tobegin, just as eq. (116) follows from eq. (28), we find that p nn +3 + γ p nn +2 + γ p nn +1 = 12 n (cid:20)(cid:18) β x ∂∂x + β y ∂∂y − γ (cid:19) p n − n +2 + (cid:18) β x ∂∂x + β y ∂∂y − γ (cid:19) p n − n +1 + (cid:18) β x ∂∂x + β y ∂∂y − γ (cid:19) p n − n (cid:21) . (143)With the definitions of eqs. (29–31), we see that eqs. (34, 117, 143) together lead to w nn +3 ( x ( t ) , y ( t ) , t ) = 12 n " − γ ddt w n − n − γ (cid:18) ddt w n − n +1 + D ( t ) w n − n (cid:19) + ddt w n − n +2 + (cid:18) β x ∂∂x + β y ∂∂y − γ (cid:19) w n − n +1 + (cid:18) β x ∂∂x + β y ∂∂y − γ (cid:19) w n − n = 12 n (cid:20) ddt w n − n +2 + D ( t ) w n − n +1 + ∆( t ) w n − n (cid:21) (144)where ∆( t ) = (cid:2) γ − γ (cid:3) (cid:20) β x ∂∂x + β y ∂∂y − γ (cid:21) − γ (cid:20) β x ∂∂x + β y ∂∂y − γ (cid:21) + (cid:20) β x ∂∂x + β y ∂∂y − γ (cid:21) . (145)Again one can iterate eq. (144) to obtain w nn +3 in terms of w , w and w as well as the two and three loop RGfunctions in the CW scheme. The summations needed to compute V N LL can then be performed using the sametechniques as were used to find V NLL in eq. (142). However, since the three loop RG functions have not beencomputed for the standard model, we will not pursue this calculation further.
Appendix 2: The Derivative Expansion of the Effective Action
This paper has been concerned with contributions to the effective action coming from the first few terms in thederivative expansion when the background field is either a scalar or vector field [16]. In this appendix we show howterms in this derivative expansion can be computed. Operator regularization [17] will be used in calculation. Thistechnique has the advantages of not explicitly breaking any classical symmetries of the theory (since no regulatingparameter is inserted into the initial action) and of avoiding all explicit divergences at every stage of the calculation.To illustrate this technique, we first consider a simple scalar model with a classical action S (0) = Z d x (cid:18) −
12 ( ∂ µ φ ) − m φ − µφ − λφ (cid:19) (146)If we split φ into the sum of a background part f and a quantum fluctuation h then performing the path integralover the quantum fluctuation leads to the one loop contribution to the effective action iS (1) = − tr ln( p + m + µf + 12 λf ) . ( p ≡ − i∂ ) (147)Regulating the logarithm in eq. (147) using the zeta function [17]ln H = − dds (cid:12)(cid:12)(cid:12)(cid:12) H − s = − dds (cid:12)(cid:12)(cid:12)(cid:12) s ) Z ∞ dit ( it ) s − e − iHt (148)we see that eq. (147) can be written iS (1) = 12 dds s ) Z ∞ dit ( it ) s − tr (cid:26) exp − i ( p + m + µf + 12 λf ) t (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) . (149)If now f → v + f where v is a constant, and if H = H + H where H = p + m + µv + 12 λv (150) H = ( µ + λv ) f + λf tr e − i ( H + H ) t = tr (cid:20) e − iH t + ( − it ) H e − iH t + 12 ( − it ) Z duH e − i (1 − u ) H t H e − iuH t + . . . (cid:21) (152)and keeping terms at most quadratic in f we obtain iS (1)2 = 12 dds κ s Γ( s ) Z ∞ dit ( it ) s − tr ( ( − it ) e − iH t (cid:20) ( µ + λv ) f + λf (cid:21) + ( − it ) Z du e − i (1 − u ) H t ( µ + λv ) f e − iuH t ( µ + λv ) f )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (153)where κ s is a dimensionful parameter inserted to ensure that S (1) is dimensionless (One could have introduced κ in eq. (147) to keep the argument of the logarithm dimensionless in that equation.).The functional trace in eq. (153) can most easily be computed using momentum eigenstates | p > , | q > andconfiguration eigenstates | x > , | y > wherein n dimensions (2 π ) x/ < x | p > = e ip · x so that iS (1)2 = 12 dds κ s Γ( s ) Z ∞ dit ( it ) s − e − i ( m + µv + λv ) t ( ( − it ) Z dpdx < p | e − ip t | x >< x | ( µ + λv ) f + λf | p > + 12 ( − it ) Z dpdqdxdy Z du < p | e − i (1 − u ) p t | x >< x | ( µ + λv ) f | q >< q | e − iuq | y >< y | ( µ + λv ) f | p > )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (154) iS (1)2 = 12 dds κ s Γ( s ) Z ∞ dit e − i ( m + µv + λv ) t ( − ( it ) s Z dpdx (2 π ) e − ip t (cid:20) ( µ + λv ) f ( x ) + λ f ( x ) (cid:21) + 12 ( it ) s +1 ( µ + λv ) Z dpdqdxdy (2 π ) e − i [(1 − u ) p + uq ] t e − i ( p − q ) · ( x − y ) f ( x ) f ( y ) )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (155)To obtain those terms which contribute to the effective action at one loop order which are second order in derivativesof the background field, we expand f ( y ) about x up to second order so that Z dpdqdxdy (2 π ) e − i [(1 − u ) p + uq ] t e − i ( p − q ) · ( x − y ) f ( x ) f ( y ) ≈ Z dpdqdxdy (2 π ) e − i [(1 − u ) p + uq ] t e − i ( p − q ) · ( x − y ) f ( x ) (cid:20) f ( x ) + ( x − y ) α f ,α ( x ) + 12 ( x − y ) α ( x − y ) β f ,αβ ( x ) (cid:21) . (156)If now we write in eq. (156) ( x − y ) α e − i ( p − q ) · ( x − y ) = − i ∂∂q α e − i ( p − q ) · ( x − y ) (157)( x − y ) α ( x − y ) β e − i ( p − q ) · ( x − y ) = ( − i ) ∂∂q α ∂∂q β e − i ( p − q ) · ( x − y ) (158)and then perform an integration by parts with respect to q we find that iS (2)2 = 12 dds κ s Γ( s ) Z ∞ dit e − i ( m + µv + λv ) t ( − ( it ) s i (4 πit ) s Z dx (cid:20) ( µ + λv ) f ( x ) + 12 λf ( x ) (cid:21) + 12 ( it ) s +1 Z du ( µ + λv ) (cid:2) f ( x ) + ( it ) u (1 − u ) f ( x ) ∂ f ( x ) (cid:3))(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (159)where we have used the integral Z d n p (2 π ) n e − ip t = i (4 πit ) n/ . (160)23he integrals over t and u are now standard and we end up with iS (1)2 = i π Z dx ((cid:20) ( µ + λv ) f ( x ) + 12 λf ( x ) (cid:21) (cid:20) m + λv + 12 λv (cid:21) (cid:20) − ln (cid:18) m + µv + λv κ (cid:19)(cid:21) −
12 ( µ + λv ) f ( x ) ln (cid:18) m + µv + λv κ (cid:19) + 12 ( µ + λv ) f ( x ) ∂ f ( x )( m + µv + λv ) ) . (161)Eq. (161) agrees with what was obtained using different techniques in ref. [16].The approach outlined for the simple scalar model of eq. (146) can easily be applied to compute terms in thederivative expansion of the effective action in more complicated models. For scalar electrodynamics with the classicalaction S φ = Z d x (cid:20) − ( ∂ µ + ieV µ ) φ ∗ ( ∂ µ − ieV µ ) φ − λ ( φ ∗ φ ) −
14 ( ∂ µ V ν − ∂ ν V µ ) (cid:21) (162)we again let φ = f + h where f is the background field. Using the gauge fixing term S gf = − α Z d x (cid:20) ∂ · V + ieα f ∗ h − f h ∗ ) (cid:21) (163)and the attendant ghost action S gh = Z d x c (cid:20) ∂ − e α (2 f ∗ f + f ∗ h + f h ∗ ) (cid:21) (164)we find that the one loop effective action is given by iS (1) = ln det (cid:2) p + e α ( f + f ) (cid:3) −
12 ln det p + 3 λf + ( λ + αe ) f (2 λ − αe ) f f − ef , ν (2 λ − αe ) f f p + ( λ + αe ) f + 3 λf ef , ν − ef ,µ ef ,µ p ( T + α L ) µν + e ( f + f ) g µν (165)where f and f are the real and imaginary parts of f and T µν = g µν − p µ p ν /p , L µν = p µ p ν /p are a complete setof orthogonal projection operators.Operator regularization can now be applied to this expression in the same way as it was applied to eq. (147);after the replacement f → v + f the Schwinger expansion is used to obtain all terms second order in f and f andthese fields can then be expanded out to second order in a Taylor expansion about some point xx