Taming the Goldstone contributions to the effective potential
aa r X i v : . [ h e p - ph ] J un Taming the Goldstone contributions to the effective potential
Stephen P. Martin
Department of Physics, Northern Illinois University, DeKalb IL 60115, andFermi National Accelerator Laboratory, P.O. Box 500, Batavia IL 60510
The standard perturbative effective potential suffers from two related prob-lems of principle involving the field-dependent Goldstone boson squared mass, G . First, in general G can be negative, and it actually is negative in the Stan-dard Model; this leads to imaginary contributions to the effective potential thatare not associated with a physical instability, and therefore spurious. Second,in the limit that G approaches zero, the effective potential minimization condi-tion is logarithmically divergent already at two-loop order, and has increasinglysevere power-law singularities at higher loop orders. I resolve both issues byresumming the Goldstone boson contributions to the effective potential. Forthe resulting resummed effective potential, the minimum value and the min-imization condition that gives the vacuum expectation value are obtained informs that do not involve G at all. Contents
I. Introduction II. Effective potential contributions for small G III. Resummation of leading Goldstone contributions IV. Minimization condition for the effective potential V. Numerical impact VI. Outlook Appendix: Effective potential results in the Standard Model References I. INTRODUCTION
A resonance with mass about 126 GeV and properties expected of a minimal StandardModel Higgs scalar boson has been discovered [1–4] at the Large Hadron Collider. One ofthe theoretical tools useful for understanding the electroweak symmetry breaking dynamicsof the minimal Standard Model and its extensions is the effective potential [5–7], whichcan be used to relate the Higgs field vacuum expectation value (VEV) to the fundamentalLagrangian parameters, and to observable quantities such as the masses of the Higgs bosons,the top quark, and the masses and interactions of the W and Z bosons. On general grounds,the Standard Model Lagrangian parameter should be obtained as accurately as possible. Itmay be possible to discern the difference between the minimal Higgs Standard Model andmore complicated theories, and to gain hints about the mass scale of new physics, and thestability of the Standard Model vacuum state. An interesting feature of the Higgs mass isthat the potential is close to a metastable region associated with a very small Higgs self-interaction coupling at very large energy scales. Some studies of the stability condition thatwere made before the Higgs discovery are refs. [7–18], and some of the analyses followingthe Higgs discovery are refs. [19–23].To fix notation, write the complex doublet Higgs field asΦ( x ) = √ [ φ + H ( x ) + iG ( x )] G + ( x ) . (1.1)Here φ is the real background field, about which are expanded the real Higgs quantum field H , and the real neutral and complex charged Goldstone boson fields G and G + = G −∗ .The Lagrangian for the Higgs kinetic term and its self-interactions are given by L = − ∂ µ Φ † ∂ µ Φ − Λ − m Φ † Φ − λ (Φ † Φ) , (1.2)where m is the Higgs squared mass parameter, and λ is the Higgs self-coupling in thenormalization to be used in this paper, and the metric signature is ( − +++). The field-independent vacuum energy density Λ is necessary for renormalization scale invariance of theeffective potential and a proper treatment of renormalization group improvement [10, 24–28].The Lagrangian also includes a top-quark Yukawa coupling y t , and SU (3) c × SU (2) L × U (1) Y gauge couplings g , g , and g ′ . The other Yukawa couplings are very small, and can makeonly a very minor difference even at 1-loop order, and so are neglected. All of the Lagrangianparameters as well as the background field φ depend on the MS renormalization scale Q ,and logarithms of dimensional quantities are written below in terms ofln( x ) ≡ ln( x/Q ) . (1.3)The effective potential can be evaluated in perturbation theory and written as: V eff ( φ ) = ∞ X ℓ =0 π ) ℓ V ℓ . (1.4)In this paper, the power of 1 / π is used as a signifier for the loop order ℓ . The tree-levelpotential is given by: V = Λ + m φ + λ φ . (1.5)The radiative corrections for ℓ ≥ V = 3 f ( G ) + f ( H ) − f ( t ) + 6 f ( W ) + W + 3 f ( Z ) + 12 Z , (1.6)where f ( x ) ≡ x (cid:2) ln( x ) − / (cid:3) , (1.7)and the field-dependent running squared masses are G = m G = m G ± = m + λφ , (1.8) H = m H = m + 3 λφ , (1.9) t = m t = y t φ / , (1.10) W = m W = g φ / , (1.11) Z = m Z = ( g + g ′ ) φ / . (1.12)The contributions W + Z in eq. (1.6) are due to the fact that in dimensional regularizationthe vector fields have 4 − ǫ components rather than 4.The full two-loop order contribution V in Landau gauge was worked out by Ford, Jack andJones in [29] for the Standard Model, and for more general theories (including softly-brokensupersymmetric models, where regularization by dimensional reduction is used instead ofdimensional regularization) in [30]. The 3-loop contribution V for the Standard Model wasobtained in [31] in the approximation that the strong and top Yukawa couplings are muchlarger than the electroweak couplings and other Yukawa couplings. For completeness, theseresults are compiled in an Appendix of the present paper in a notation compatible with thediscussion below.Results for more general gauge-fixing conditions are apparently only available at 1-looporder at present. The effective potential itself is gauge-fixing dependent, but physical ob-servables derived from it are not. For discussions of the gauge-fixing dependence of theeffective potential from various points of view see refs. [32–47].The purpose of this paper is to resolve two issues of principle regarding contributionsto the effective potential involving the Goldstone bosons. Note that the condition G = 0marks the minimum of the tree-level potential V in eq. (1.5). However, in general G will benon-zero at the minimum of the full effective potential.The first problem of principle is that there is no reason why G cannot be negative at theminimum of V eff , depending on the choice of renormalization scale Q . Indeed, in the case ofthe Standard Model, G is negative for the perfectly reasonable range Q ∼ >
100 GeV. This isillustrated in Figure 1.1 for a typical numerical choice of the parameters (in this case takenfrom ref. [23]): λ ( M t ) = 0 . , (1.13) y t ( M t ) = 0 . , (1.14) g ( M t ) = 1 . , (1.15) m ( M t ) = − (93.36 GeV) , (1.16) g ( M t ) = 0 . , (1.17) g ′ ( M t ) = 0 . . (1.18)where Q = M t = 173 .
35 GeV is the input scale. The problem is that the ln( G ) terms giverise to an imaginary part of V eff . In general, a complex value of V eff at its minimum reflectsan instability [48], but there is no physical instability here. In practice, this unphysicalimaginary part of V eff has simply been ignored, and the real part is minimized. In principle,this imaginary part is spurious, and a way of making this plain is desirable.The second problem of principle is that when calculated to fixed loop order ℓ , the effective
50 100 150 200 250 300
Renormalization scale Q [GeV] -(40) -(30) -(20) (30) (40) G = t r ee - l e v e l G o l d s t one m a ss [ G e V ] At Q = 173.35 GeV: v = 246.954 GeV λ = 0.12710y t = 0.93697g = 1.1666g = 0.6483g ’ = 0.3587m = -(93.36 GeV) FIG. 1.1: The running of the Lan-dau gauge Standard Model Goldstoneboson squared mass G , evaluated atthe minimum of the effective poten-tial, as a function of the renormaliza-tion scale Q , for the choice of inputparameters specified in the text. potential diverges for G →
0. For ℓ = 1 , ,
3, one finds [31] that V ℓ ∼ G − ℓ ln G , whilefor ℓ ≥ V ℓ ∼ G − ℓ . (The precise results for the leading behavior as G → ℓ = 3, is quite small unless one carefully tunes G ≈ Q that makes G very small would be a particularly good choice, because thepole mass of the Landau gauge Goldstone bosons is 0, and so G = 0 corresponds to choosinga renormalization scale such that the radiative corrections to it vanish. However, if one usesthe usual perturbative effective potential truncated at any loop order beyond 1-loop order,this is the one renormalization scale choice that one must not make.The purpose of this paper is to show how the above two problems of principle are elim-inated by doing a resummation to all loop orders of the Goldstone contributions that areleading as G → II. EFFECTIVE POTENTIAL CONTRIBUTIONS FOR SMALL G For the known contributions to the effective potential, the leading behavior as G → G , G ± G tt G ± bt G G tt tt G ± G ± bb tt FIG. 2.1: These Feynman diagrams give the leading non-zero contribution to V eff as G →
0, at theleading order in y t , for 1-loop, 2-loop, and 3-loop orders. the Goldstone boson contributions are: V = 3 f ( G ) + . . . = 34 G [ln( G ) − /
2] + . . . . (2.1)In eq. (2.1), the ellipses represent terms with no G dependence. At 2-loop order, using theexpansions of the 2-loop integral function I ( x, y, z ) for small G given in eqs. (A.26)-(A.29)of the Appendix (from eqs. (2.29)-(2.31) of ref. [30]), one finds that V = 32 ∆ G ln( G ) + . . . , (2.2)where ∆ = − y t A ( t ) + 3 λA ( H ) + g A ( W ) + 2 W ] + g + g ′ A ( Z ) + 2 Z ] , (2.3)and the 1-loop integral function is defined by A ( x ) ≡ x (ln x − . (2.4)(In ref. [30], A ( x ) was called J ( x ).) In eq. (2.2), the ellipses represent terms independent of G , terms of order G , as well as those proportional to G with no ln( G ). Reading directlyfrom eq. (4.38) of ref. [31], one obtains the leading behavior as G → V = 27 y t A ( t ) ln( G ) + . . . . (2.5)As noted in ref. [31], the contributions above (that involve y t , in the 2-loop case) come fromthe diagrams shown in Figure 2.1. At higher loop orders, the leading contribution as G → y t comes from ℓ -loop order vacuum diagrams consisting of a ring of ℓ − ℓ − G ) or top/bottom (for G ± ) 1-loop subdiagrams, as shown in Figure 2.2.More generally still, the 1-loop subdiagrams can be any 1-particle irreducible sub- G G G G tt tttt G ± G ± G ± G ± tb tbtb FIG. 2.2: Chains of Goldstone boson propagators interspersed with top and top/bottom loops.Rings of these (and similar diagrams involving loops with Z , W , and H ) give rise to the mostsingular contributions as G →
0, at any given loop order. diagrams that involve a squared mass scale that can be treated as parametrically largecompared to G . In the Standard Model, these includes 1-loop subdiagrams containing Z , W , and H bosons, as well as multi-loop subdiagrams.To obtain the leading behavior as G →
0, one considers the contributions from thesediagrams from integrating momenta p µ flowing around the large rings with p small com-pared to the squared mass scale set by the 1-particle-irreducible sub-diagrams. Then the1-particle-irreducible sub-diagram contributions can be treated as just constant squared-mass insertions in the Goldstone boson propagators, reducing the calculation to a 1-loopintegration. The sum of the resulting leading G → V eff at each loop order,including the MS counterterms, is: V eff = 316 π ∞ X n =0 ∆ n n ! (cid:18) ddG (cid:19) n f ( G ) + . . . , (2.6)where n = ℓ − ℓ is the loop order, and∆ = 116 π ∆ + 1(16 π ) ∆ + 1(16 π ) ∆ + . . . . (2.7)Equation (2.6) generalizes eqs. (2.1), (2.2), and (2.5). For the purposes of making thisparticular comparison, ∆ can be dropped, because ref. [31] retained only the leading orderin y t at 3 loops. For the same reason, all but the y t term in ∆ can be dropped in comparingthe 3-loop order contributions. However, in the future, if more terms are calculated in V eff at 3-loop order and beyond, then those contributions would become pertinent, as wouldcontributions from other diagrams.The origin of the prefactor 3 in eq. (2.6) is a factor of 2 for the G ± rings, and a factor1 for the G ring. Despite the fact that the 1-particle-irreducible subdiagrams are differentfor these two classes of diagrams (e.g. involving top/bottom loops for G ± , and top loops for G ), the quantity ∆ is the same in both cases.Note that f ′ ( G ) = A ( G ) = G [ln( G ) − f ′′ ( G ) = ln( G ), and the n th derivativeis f ( n ) ( G ) = ( − n − ( n − G − n for n ≥
3. Therefore, the leading singular behavior as G → V = 34 (∆ ) ln( G ) , (2.8) V ℓ = − − ∆ ) ℓ − ℓ − ℓ − ℓ − G ℓ − (for ℓ > III. RESUMMATION OF LEADING GOLDSTONE CONTRIBUTIONS
The contributions to V eff in eq. (2.6) from all loop orders ℓ = n + 1 resum to V eff = 316 π f ( G + ∆) + . . . . (3.1)Thus, the result of summing all orders in perturbation theory yields a result which is well-behaved for all G , unlike the result obtained if it is truncated at any finite order in perturba-tion theory. In fact, at the minimum of the full effective potential, G + ∆ = 0, and the resultof the resummation of this class of terms vanishes. Therefore, if V eff has been evaluated atsome finite ℓ -loop order in perturbation theory, a sensible result can be obtained by simplysubtracting off the first ℓ terms in the series eq. (2.6), and then adding back in the resummedversion of the same series, eq. (3.1). If the effective potential V eff has been calculated to looporder ℓ , then the resummed effective potential is b V eff = V eff + 316 π " f ( G + ∆) − ℓ − X n =0 ∆ n n ! (cid:18) ddG (cid:19) n f ( G ) . (3.2)The result is free of the offending leading singular contributions as G → G → V eff to refer to the usual full 2-loopand leading 3-loop Standard Model effective potential as computed in refs. [29] and [31],then the appropriate resummed version from eq. (3.2) is: b V eff = V eff + 316 π [ f ( G + ∆) − f ( G )] − π ) A ( G ) − π ) y t A ( t ) ln( G ) . (3.3)Here I have taken (∆ ) = 36 y t A ( t ) in the 3-loop part, because only the leading order in y t for V eff was included in ref. [31]. For the same reason, the 3-loop order term involving ∆ is dropped here. The effect of the 1-loop correction in eq. (3.3) is to replace the tree-levelfield-dependent Goldstone boson squared mass by its pole squared mass, which vanishes atthe minimum of the full effective potential in Landau gauge. The 3-loop order term simplycancels the corresponding ln( G ) contribution found in ref. [31]. I propose that the resummedversion of the effective potential, b V eff , should be used instead of the usual V eff . IV. MINIMIZATION CONDITION FOR THE EFFECTIVE POTENTIAL
For the usual effective potential V eff , the minimization condition that relates the vacuumexpectation value v = φ min to the Lagrangian parameters is G = m + λv = − π δ − π ) δ − π ) δ − . . . , (4.1)where the correction at ℓ -loop order is: δ ℓ = 1 φ ∂∂φ V ℓ (cid:12)(cid:12)(cid:12) φ = v . (4.2)(In the remainder of this section, I will use φ = v , because all equations hold only at theminimum of the potential.) Explicitly, at 1-loop order, one has from eq. (1.6): δ = 3 λA ( G ) − y t A ( t ) + 3 λA ( H ) + g A ( W ) + 2 W ] + g + g ′ A ( Z ) + 2 Z ] , (4.3)and the higher loop order contributions can be obtained by taking derivatives of the resultsof ref. [29] at 2-loop order, and from ref. [31] at 3-loop order for terms that are leading orderin g and y t . Note that δ differs from the quantity ∆ , given in eq. (2.3) above, only bythe inclusion of the first term, 3 λA ( G ). At higher loop orders ℓ ≥
2, it is useful to note theleading dependence on G as G → δ = 3 λ ∆ ln( G ) − y t ln( t ) A ( G ) + . . . , (4.4) δ = 54 y t (cid:20) λA ( t ) G + y t A ( t )ln( t )ln( G ) (cid:21) + . . . . (4.5)Equation (4.4) can be obtained using eqs. (A.1)-(A.21) and (A.26)-(A.29) in the Appendix,and eq. (4.5) can be obtained from eq. (A.30). In δ , the ellipses includes terms that do notdepend on G , terms with a linear factor of G but suppressed by g, g ′ , λ or not containingln( G ), and terms quadratic or higher order in G . In δ , the ellipses represents terms thatare finite as G → g, g ′ , λ .0The condition for minimization of b V eff defined in eq. (3.3) is: G = m + λv = − π ∆ − π ) ∆ − π ) ∆ − . . . , (4.6)where ∆ ℓ = 1 v ∂∂v b V ℓ , (4.7)with b V eff = X ℓ π ) ℓ b V ℓ . Consider the difference between the minimization conditions for V eff and b V eff . First, note that the term proportional to f ( G + ∆) gives no contribution,because f ′ ( G + ∆) = ( G + ∆)(ln( G + ∆) −
1) vanishes at the minimum of the potential,where G + ∆ = 0. One therefore finds from eq. (3.3):1 v ∂∂v (cid:16) b V eff − V eff (cid:17) = − π λA ( G ) − π ) (cid:20) λ ∆ ln( G ) + 32 A ( G ) 1 v ∂ ∆ ∂v (cid:21) − π ) y t (cid:20) λA ( t ) G + y t A ( t )ln( t )ln( G ) (cid:21) . (4.8)Note that the apparently 2-loop term containing A ( G ) is actually of 3-loop order, because A ( G ) contains a factor of G , which at the minimum of the potential is equal to − π ∆ + . . . .Therefore, in eq. (4.8), to be consistent we should keep only the part of v ∂ ∆ ∂v that involvesthe top Yukawa coupling: 1 v ∂ ∆ ∂v = − y t ln( t ) + . . . . (4.9)So, one obtains from eq. (4.8):∆ = δ − λA ( G ) , (4.10)∆ = δ − λ ∆ ln( G ) + 9 y t ln( t ) A ( G ) , (4.11)∆ = δ − y t (cid:20) λA ( t ) G + y t A ( t )ln( t )ln( G ) (cid:21) . (4.12)This shows that the G -dependent terms of eqs. (4.3)-(4.5) neatly cancel, up to the orderthat has been calculated.The resulting ∆ does not explicitly depend on G , but depends on m through H . A1further refinement of the minimization condition can be made by writing H = h + G, (4.13)where, from eq. (1.9), h = 2 λv (4.14)at the minimum of the potential, and then iteratively replacing the G dependence using G n ln p ( G ) = (cid:18) − π ∆ − π ) ∆ − . . . (cid:19) n ln p ( G ) (4.15)in ∆ , ∆ , and ∆ . (Note that logarithms of G are left alone, to cancel amongst themselves.)In doing so, one consistently drops terms of 4-loop order as well as terms of 3-loop orderthat are suppressed by g , g ′ or λ . Thus, for example, the 1-loop contribution involving H isrewritten using A ( H ) = A ( h ) + GA ′ ( h ) + G A ′′ ( h ) + . . . (4.16)= A ( h ) − π ∆ ln( h ) + 1(16 π ) (cid:20) (∆ ) h − ∆ ln( h ) (cid:21) + . . . . (4.17)Because this is multiplied by 3 λ/ π in the minimization condition, the ∆ term should nowbe dropped, but the (∆ ) / h term is partially kept, because 3 λ (∆ ) / h = 27 y t A ( t ) /v + . . . . The self-consistent elimination of G from the right side of the minimization conditionshifts contributions that were originally proportional to G n up in loop order by n , wherethey can often therefore be dismissed. (Note that this iterative elimination of G from theright-hand side of the minimization condition would not have been possible without firsteliminating by resummation the terms that behave like ln( G ) and 1 /G as G → b V eff : G = m + λv = − π b ∆ − π ) b ∆ − π ) b ∆ − . . . , (4.18)where the b ∆ ℓ depend on the VEV v and the couplings λ, y t , g , g, g ′ , or on h, t, W, Z , but donot depend at all on G or m . The results are: b ∆ = − y t A ( t ) + 3 λA ( h ) + g A ( W ) + 2 W ] + g + g ′ A ( Z ) + 2 Z ] , (4.19)2 b ∆ = (3 g − g ′ )(33 g + 22 g g ′ + g ′ )8( g + g ′ ) I ( W, W, Z )+ (cid:20) (9 g + 66 g g ′ − g ′ ) y t g + g ′ ) − g + 12 g g ′ − g ′ (cid:21) I ( t, t, Z )+ (cid:20) g + g ′ ) g + g ′ − λ ) − g + g ′ ) + 114 λ ( g + g ′ ) − λ (cid:21) I ( h, Z, Z )+ (cid:20) g g − λ ) − g + 112 λg − λ (cid:21) I ( h, W, W )+ y t (cid:20) y t − λ (cid:21) I ( h, t, t ) − λy t I (0 , , t ) − λ I ( h, h, h ) − λ I (0 , , h ) + 34 λ ( g + g ′ + 8 λ ) I (0 , h, Z ) + 32 λ ( g + 8 λ ) I (0 , h, W )+ 3 λ (2 g + g ′ ) g + g ′ ) I (0 , W, Z ) + (6 y t + 3 g y t − g ) I (0 , t, W )+ (cid:20) λg − g − λg (3 g + 2 g ′ )2( g + g ′ ) (cid:21) I (0 , , W )+ (cid:20) λg g + g ′ ) − g λ − g − g g ′ − g ′ (cid:21) I (0 , , Z )+ (cid:20) λ − g + g ′ g + g ′ ) λ − g − g ′ ) (cid:21) A ( Z ) /v + (cid:20) λ − g + g ′ g g + g ′ + 3 g λ − g ) (cid:21) A ( W ) /v + (cid:20) λ (8 g + 8 g g ′ + g ′ )( g + g ′ ) + 15 g − g g ′ − g ′ g + g ′ (cid:21) A ( W ) A ( Z ) /v − g − g g ′ + 17 g ′ )3( g + g ′ ) A ( t ) A ( Z ) /v + 12( y t − g ) A ( t ) A ( W ) /v + (cid:20) g + 15 y t + 6 λ + 9 g + 90 g g ′ + 17 g ′ g + g ′ ) (cid:21) A ( t ) /v + (cid:20)
34 ( g + g ′ ) − λ + 3( g + g ′ ) g + g ′ − λ ) (cid:21) A ( h ) A ( Z ) /v + (cid:20) g − λ + 3 g g − λ ) (cid:21) A ( h ) A ( W ) /v − y t A ( h ) A ( t ) /v + (cid:20) g + g ′ ) λ − g − g ′ ) + λ ( g ′ − g )2 − y t (63 g + 30 g g ′ + 95 g ′ )12( g + g ′ )+ 6 g g + g ′ + 2948 g + 4 g g ′ + 45548 g ′ (cid:21) A ( Z )+ g (cid:20) g λ − g ) − y t − g ( λ + 4 g ) g + g ′ + 60524 g + 138 g ′ − λ (cid:21) A ( W )+ y t (cid:20) g − y t + 24 λ − g + 113 g ′ + 64 g g + g ′ ) (cid:21) A ( t )3+ (cid:20) g + g ′ ) λ − g − g ′ ) + 3 g λ − g ) − y t + 9 λy t + 24 λ − λ (3 g + g ′ ) + 92 g + 3 g g ′ + 32 g ′ (cid:21) A ( h )+ (cid:20) g y t + 9 y t + 6 y t λ + y t (18 g + 87 g g ′ + 5 g ′ )6( g + g ′ ) + 9 g g − λ )+ 9( g + g ′ ) g + g ′ − λ ) + 3 g (8 g + λ )8( g + g ′ ) − λ + λ (6 g + 2 g ′ − y t ) − y t g ′ + 238 y t g g ′ − y t g + 9316 λg + 78 λg g ′ + λg ′
16+ 19964 g − g g ′ − g g ′ − g ′ (cid:21) v , (4.20) b ∆ = g y t v h . − .
20 ln( t ) + 592 ln ( t ) −
184 ln ( t ) i + g y t v h − .
84 + 860 .
93 ln( t ) −
270 ln ( t ) + 60 ln ( t ) i + y t v (cid:20) − . − .
02 ln( t ) + 36 ln( h )ln( t ) + 6578 ln ( t )+54 ln( h )ln ( t ) − ( t ) (cid:21) + . . . . (4.21)Here, the ellipses represent terms suppressed by g , g ′ , or λ . The analytical versions of thedecimal coefficients in eq. (4.21) are:1036 . ≈ ζ (3) + 176135 π + 649 ln (2)[ π − ln (2)] − (1 / , (4.22) − . ≈ ζ (3) − / , (4.23) − . ≈ − − π − ζ (3) + 12415 π + 1283 ln (2)[ π − ln (2)] − (1 / , (4.24)860 . ≈
454 + 12 π + 240 ζ (3) , (4.25) − . ≈ π − ζ (3) − π − (2)[ π − ln (2)] + 192 Li (1 / , (4.26) − . ≈ − − π − ζ (3) . (4.27)Having found the minimization condition in a form that does not depend on G , one cannow write the value of b V eff at its minimum, again eliminating all G and m dependence bythe same procedure. The result is: b V eff , min = ∞ X ℓ =0 π ) ℓ b V ℓ, min , (4.28)4where b V , min = Λ − λv / , (4.29) b V , min = 3 t (cid:2) ln( t ) − / (cid:3) − h (cid:2) ln( h ) − / (cid:3) − W (cid:2) ln( W ) + 1 / (cid:3) − Z (cid:2) ln( Z ) + 1 / (cid:3) , (4.30) b V , min = − λ v [ I ( h, h, h ) + I ( h, , − λA ( h ) +3 y t h (2 t − h/ I ( h, t, t ) + tI (0 , , t ) + A ( t ) i + g + g ′ f SSV (0 , h, Z ) + ( g − g ′ ) g + g ′ ) f SSV (0 , , Z ) + g f SSV (0 , h, W )+ g f SSV (0 , , W ) + g g ′ v g + g ′ ) (cid:2) g ′ f V V S ( W, Z,
0) + g f V V S (0 , W, (cid:3) + g v f V V S ( W, W, h ) + ( g + g ′ ) v f V V S ( Z, Z, h )+ V F F V + V gauge − v b ∆ , (4.31) b V , min = g t n ( t ) − ( t ) + h − ζ (3) i ln( t ) + 2939 − ζ (3) − π −
649 ln (2)[ π − ln (2)] + 5123 Li (1 / o + g y t t n − ( t ) + 180ln ( t ) + (cid:2) − − π − ζ (3) (cid:3) ln( t ) − π − ζ (3) − π − (2)[ π − ln (2)] + 1024Li (1 / o + y t t n ( t ) + h − − h ) i ln ( t )+ h π + 36 ζ (3) − h ) i ln( t ) − − π + 2215 π + 9452 ζ (3) + 8 ln (2)[ π − ln (2)] − (1 / o , (4.32)Here v, h, t, W, Z are understood to be evaluated at the solution of the minimization con-dition, given by eqs. (4.18)-(4.21). Although eqs. (4.28)-(4.32) have only been computedin Landau gauge here, the value of the effective potential at its minimum is in principle aphysical observable and does not depend on the choice of gauge fixing, unlike the VEV itself.Renormalization group scale invariance provides an important and non-trivial check onthe results above. If one acts on each side of eq. (4.18) with Q ddQ = Q ∂∂Q − γ φ v ∂∂v + X X β X ∂∂X , (4.33)5where X = { λ, y t , g , g, g ′ , m } , then the results must match, up to terms of 3-loop ordersuppressed by λ, g, g ′ and terms of 4-loop order. I have checked this, using the beta functionsand scalar field anomalous dimension γ φ given at the pertinent orders in eqs. (A.32)-(A.45) inthe Appendix. Equations (A.23)-(A.25) are also useful in conducting this check. Similarly,I have checked that acting with eq. (4.33) on eq. (4.28) gives 0, up to terms of 3-loop ordersuppressed by λ, g, g ′ and terms of 4-loop order, as required. In that check, one has instead X = { λ, y t , g , g, g ′ , Λ } , with the beta function for the field-independent vacuum energydensity given by eqs. (A.46)-(A.47).To conclude this section, I remark on a different expansion procedure that eliminatesthe Goldstone boson dependence of the minimization condition for the effective potential,proposed in ref. [45] to avoid spurious gauge dependence in physical quantities in the con-text of a more general gauge-fixing and finite temperature field theory with applications tobaryogenesis. The idea, called the “¯ h expansion” in ref. [45], is to first write an expansionof the VEV in the same way as the effective potential: v = φ min = φ + 116 π φ + 1(16 π ) φ + 1(16 π ) φ + . . . (4.34)and then to demand that, after expanding in the loop-counting parameter (here 1 / π ),the contributions to the derivative of V eff vanish separately at each loop order. Then φ minimizes the tree-level potential V , so that m + λφ = 0 and V ′′ ( φ ) = 2 λφ , with thesubsequent terms in the expansion of the VEV given by: φ = − V ′ /V ′′ , (4.35) φ = − h V ′ + φ V ′′ + 12 φ V ′′′ i /V ′′ , (4.36) φ = − h V ′ + φ V ′′ + φ V ′′ + 12 φ V ′′′ + φ φ V ′′′ + 16 φ V ′′′′ i /V ′′ , (4.37)etc., with all of the derivatives of V ℓ on the right-hand sides evaluated at the tree-levelminimum, φ = p − m /λ . Each of the individual terms V ′′ , V ′′′ , V ′ , V ′′ , and V ′ divergesas φ → φ , but one can check that the combination φ is well-behaved in this limit, andthat φ is well-behaved up to the approximation corresponding to the known calculationof V in ref. [31]. The result is therefore indeed free of spurious imaginary parts and the G → y t is much larger than λ , and g, g ′ = 0: φ = 3 y t λ (cid:2) ln( t ) − (cid:3) φ + . . . , (4.38) φ = 9 y t λ (cid:2) ln( t ) − (cid:3) (cid:2) t ) (cid:3) φ + . . . , (4.39) φ = 27 y t λ (cid:2) ln( t ) − (cid:3) h − t ) + 5ln ( t ) i φ + . . . , (4.40)with t = y t φ /
2. The presence of powers of λ in the denominators is due to V ′′ = 2 λφ .This shows that, due to expanding around the tree-level VEV, the expansion parameteris effectively y t π λ , rather than the usual perturbative expansion parameters y t π and λ π . Correspondingly, in this approach some of the information present in the known V , V , V , and V evaluated at the minimum of the full effective potential is postponed to thecontributions φ ℓ with ℓ ≥
4. With the presently known approximation to V found in ref. [31],the finiteness of φ [as the limit φ → φ is taken in the derivatives of V ℓ in eq. (4.37)] onlyworks up to the order y t φ /λ (in an expansion of φ in small λ/y t ). A further calculation ofsubleading corrections to V in the expansion in λ would be necessary to make well-definedthe φ contributions of order y t φ and g y t φ . Despite these formal issues, I have checkedthat in practice the numerical result of applying the expansion procedure of ref. [45] as ineqs. (4.34)-(4.37) above, with all known effective potential contributions included, agreesvery well with the expansion found in the present paper, eqs. (4.18)-(4.21). For the inputparameters of eqs. (1.13)-(1.18), the two methods agree on the predicted VEV to within 20MeV. V. NUMERICAL IMPACT
The numerical effect of the resummation is very small for almost all choices of the renor-malization scale. This is illustrated in Figure 5.1. In the left panel, the input parametersare specified by eqs. (1.13)-(1.18) at the input scale Q = 173 .
35 GeV. These are then runusing the full 3-loop renormalization group equations [50]-[56] to the scale Q , where theminimization of the effective potential gives the VEV v . Three different approximations areshown: the full 2-loop order V eff of ref. [29], the same result including partial 3-loop ordercontributions from [31], and the result after resummation, using eqs. (4.18)-(4.21). (Forthe first two approximations, the effective potential is complex for all Q ∼ > . v ,near the renormalization scale at which G crosses through 0 (compare Figure 1.1). This is7
50 100 150 200 250 300
Renormalization scale Q [GeV] v = v a c uu m e x pe c t a t i on v a l ue [ G e V ] At Q = 173.35 GeV:m = -(93.36 GeV) λ = 0.12710y t = 0.93697g = 1.1666g = 0.6483g ’ = 0.3587
50 100 150 200 250 300
Renormalization scale Q [GeV] (- m ) / [ G e V ] At Q = 173.35 GeV:v = 246.954 GeV λ = 0.12710y t = 0.93697g = 1.1666g = 0.6483g ’ = 0.3587 FIG. 5.1: Dependence of the VEV v (left panel) and the Higgs Lagrangian mass parameter √− m (right panel), as a function of the renormalization scale, as computed from the effective potentialminimization condition at 2-loop order from ref. [29], at partial 3-loop order including also [31],and 3-loop order after resummation using eqs. (4.18)-(4.21). The input parameters are specified ineqs. (1.13)-(1.18) at the input scale Q = 173 .
35 GeV. In the left panel, the parameters including m are run from the input scale to Q , and v is solved for. In the right panel, the parameters (including v = 246 .
954 GeV at Q = 173 .
35 GeV) are run to Q , and m is solved for. The 3-loop case withoutresummation has a numerical instability associated with a failure of the iterative solution processto converge, represented by the vertical line of arbitrary height, for a narrow range near Q = 100 . /G term in the minimization condition. represented by a vertical line in the figure, for a range of several hundred MeV in Q near Q = 100 . Q , the 3-loop result for v islower than the 2-loop result by up to a few hundred MeV, depending on Q . The resummed3-loop result does not differ much from the non-resummed result, except for the numericalinstability region just mentioned. There, the resummed result of course remains perfectlysmooth, as it does not depend on G at all.The right panel of Figure 5.1 shows the result of running the same parameters but v instead of m , starting again from the input scale at 173.35 GeV, and then solving for m at the scale Q using eqs. (4.18)-(4.21). Again the difference between the usual 3-loop andresummed 3-loop calculations is very small except near the scale Q = 100 . G goes through 0, where the iterative process of solution again fails for the non-resummedcase.8 VI. OUTLOOK
In this paper, I have shown how issues of principle associated with the Goldstone bosoncontributions to the effective potential can be resolved through resummation. The minimiza-tion condition of eqs. (4.18)-(4.21), and the value of the effective potential at its minimumeqs. (4.28)-(4.32), do not involve G at all, and so are manifestly free of spurious imaginaryparts associated with ln( G ) when G is negative, and of divergences as G → Q to ensure that G is not too close to 0. Here, I note that the minimization conditions eqs. (4.18)-(4.21) areactually easier to implement in practice, because there are fewer and less complicated termsand no need to deal with imaginary parts. The resummation method described above is alsoa useful ingredient in the analytical 2-loop calculation of the Standard Model Higgs mass.Both the resummed version of the minimization condition eqs. (4.18)-(4.21) and the 2-loopHiggs mass in the Standard Model will be implemented in a forthcoming public computercode [57].A similar resummation of Goldstone contributions can clearly be applied to other cases ofsymmetry breaking beyond the Standard Model. For supersymmetry, the second derivativesof V eff have been used in one of the methods for approximating the lightest Higgs boson mass.The effective potential in minimal supersymmetry has the same behavior [58, 59] with respectto the tree-level squared masses of the Goldstone bosons at 2-loop order (except that m G ± and m G are slightly different from each other when not at the minimum of the tree-levelpotential). The use of second derivatives, rather than first derivatives as in the minimizationcondition, means that the numerical instabilities associated with choices of renormalizationscale where m G ± ≈ m G ≈ Appendix: Effective potential results in the Standard Model
The 2-loop contribution to the Landau gauge Standard Model effective potential wasfound in ref. [29]. The result is: V = V SSS + V SS + V F F S + V SSV + V V V S + V V S + V F F V + V gauge , (A.1)where V SSS = − λ v [ I ( H, H, H ) + I ( H, G, G )] , (A.2) V SS = 34 λ (cid:2) A ( H ) + 2 A ( H ) A ( G ) + 5 A ( G ) (cid:3) , (A.3) V F F S = 3 y t h (2 t − H/ I ( H, t, t ) − G I ( G, t, t ) + ( t − G ) I (0 , G, t )+ A ( t ) − A ( H ) A ( t ) − A ( G ) A ( t ) i , (A.4) V SSV = g + g ′ f SSV ( G, H, Z ) + ( g − g ′ ) g + g ′ ) f SSV ( G, G, Z )+ g g ′ g + g ′ ) f SSV ( G, G,
0) + g f SSV ( G, H, W ) + f SSV ( G, G, W )] , (A.5) V V V S = g g ′ v g + g ′ ) (cid:2) g ′ f V V S ( W, Z, G ) + g f V V S (0 , W, G ) (cid:3) + g v f V V S ( W, W, H ) + ( g + g ′ ) v f V V S ( Z, Z, H ) , (A.6) V V S = ( g − g ′ ) g + g ′ ) f V S ( Z, G ) + g + g ′ f V S ( Z, H ) + f V S ( Z, G )]+ g f V S ( W, H ) + 3 f V S ( W, G )] , (A.7) V F F V = − (cid:20) g + 4 g g ′ g + g ′ ) (cid:21) tf F F V ( t, t,
0) + 3 g f F F V (0 , t, W ) + 3 f F F V (0 , , W )]+ 124( g + g ′ ) (cid:20) (9 g − g g ′ + 17 g ′ ) f F F V ( t, t, Z )+8 g ′ (3 g − g ′ ) tf F F V ( t, t, Z ) + (63 g + 6 g g ′ + 103 g ′ ) f F F V (0 , , Z ) (cid:21) , (A.8) V gauge = g g ′ g + g ′ ) f gauge ( W, W,
0) + g g + g ′ ) f gauge ( W, W, Z ) . (A.9)Here the loop functions are written in terms of the 1-loop function A ( x ) defined in eq. (2.4)and a 2-loop function I ( x, y, z ), which is invariant under interchange of any pair of x, y, z .It is equal to the ǫ -independent part of the function (16 π ) ˆ I ( x, y, z ) defined in ref. [29], andit is also given in terms of dilogarithms in eq. (2.19) of [30], and defined in section 2 of [49],0which also provides a public computer code that evaluates it efficiently. The special cases ofthe functions f SSV , f V V S , f V S , f F F V , f F F V , and f gauge defined in ref. [30] that are pertinentfor the Standard Model are: f SSV ( x, y, z ) = (cid:2) − ( x + y + z − xy − xz − yz ) I ( x, y, z ) + ( x − y ) I (0 , x, y )+( x − y − z ) A ( y ) A ( z ) + ( y − x − z ) A ( x ) A ( z ) (cid:3) /z + A ( x ) A ( y ) + 2( x + y − z/ A ( z ) , (A.10) f SSV ( x, x,
0) = − A ( x ) + 8 xA ( x ) − x , (A.11) f V V S ( x, y, z ) = (cid:2) − ( x + y + z + 10 xy − xz − yz ) I ( x, y, z )+( x − z ) I (0 , x, z ) + ( y − z ) I (0 , y, z ) − z I (0 , , z )+ yA ( x ) A ( z ) + xA ( y ) A ( z ) + ( z − x − y ) A ( x ) A ( y ) (cid:3) / xy + A ( x ) / A ( y ) / A ( z ) − x − y − z, (A.12) f V V S (0 , x, y ) = [(3 y − x ) I (0 , x, y ) − yI (0 , , y ) + 3 A ( x ) A ( y )] / x +2 A ( y ) − x/ − y/ , (A.13) f V S ( x, y ) = 3 A ( x ) A ( y ) + 2 xA ( y ) , (A.14) f F F V ( x, x,
0) = 0 , (A.15) f F F V ( x, x,
0) = 4 A ( x ) − x − A ( x ) /x, (A.16) f F F V (0 , x, y ) = (cid:2) ( x − y )( x + 2 y ) I (0 , x, y ) − x I (0 , , x ) + ( x − y ) A ( x ) A ( y ) (cid:3) /y +(2 y/ − x ) A ( y ) − xA ( x ) + x − y , (A.17) f F F V ( x, x, y ) = 2( x − y ) I ( x, x, y ) + 2 A ( x ) − A ( x ) A ( y ) − xA ( x )+ (2 y/ − x ) A ( y ) + 4 x − y , (A.18) f F F V ( x, x, y ) = 6 I ( x, x, y ) − A ( x ) + 4 x + 2 y, (A.19) f gauge ( x, x, y ) = (4 x − y )(12 x + 20 xy + y ) I ( x, x, y ) / x +( x − y ) ( x + 10 xy + y ) I (0 , x, y ) / x y + y (2 x − y ) I (0 , , y ) / x + x (2 y − x ) I (0 , , x ) / y +[ y + 18 xy − x ] A ( x ) / x + [ x + 23 xy − y ] A ( x ) A ( y ) / xy +[11 y + 25 x/ A ( x ) + [11 x − y/ A ( y ) + 14 x + 24 xy + y , (A.20) f gauge ( x, x,
0) = 9 x I (0 , , x ) − A ( x ) + 100 x A ( x ) − x . (A.21)In this paper, the explicit form of I ( x, y, z ) is not needed; instead, the calculations rely onseveral identities that it satisfies. First, we have I (0 , x, x ) = 2 A ( x ) − x − A ( x ) /x, (A.22)1which has been used in writing the equations above. Derivatives with respect to squaredmass arguments are ∂∂x I ( x, y, z ) = h ( x − y − z ) I ( x, y, z ) − A ( y ) A ( z ) + ( x − y + z ) A ( x ) A ( y ) /x +( x + y − z ) A ( x ) A ( z ) /x + ( y + z − x )[ A ( x ) + A ( y ) + A ( z )]+ x − ( y + z ) i / ( x + y + z − xy − xz − yz ) , (A.23) ∂∂x I (0 , x, x ) = − A ( x ) /x . (A.24)The derivative with respect to the renormalization scale Q is: Q ∂∂Q I ( x, y, z ) = 2 A ( x ) + 2 A ( y ) + 2 A ( z ) − x − y − z. (A.25)For making expansions in small G , the following results from eqs. (2.29)-(2.31) of ref. [30]are useful: I (0 , , G ) = G (cid:20) −
12 ln ( G ) + 2ln( G ) − − π (cid:21) , (A.26) I ( G, G, x ) = I (0 , , x ) + 2 G (cid:2) − x − I (0 , , x ) + 3 A ( x ) − A ( x )ln( G ) (cid:3) /x + O ( G ) , (A.27) I ( G, x, y ) = I (0 , x, y ) + G h − ( x + y ) I (0 , x, y ) − A ( x ) A ( y )+3 xA ( x ) + 3 yA ( y ) − yA ( x ) − xA ( y ) − ( x + y ) +( x − y )[ A ( y ) − A ( x )]ln( G ) i / ( x − y ) + O ( G ) , (A.28) I ( G, x, x ) = 2 A ( x ) − x − A ( x ) /x + G (cid:2) A ( x ) / (2 x ) + 3 A ( x ) /x − [1 + A ( x ) /x ]ln( G ) (cid:3) + O ( G ) . (A.29)The 3-loop contribution to the Standard Model effective potential, in the approximation g , y t ≫ λ, g, g ′ , was found in ref. [31] (where it was written slightly differently): V = g t n − ( t ) + 868ln ( t ) + (32 ζ (3) − t ) + 166339+32 ζ (3) + 176135 π + 649 ln (2)[ π − ln (2)] − (1 / o + g y t t n ( t ) − ( t ) + (cid:2)
814 + 12 π + 240 ζ (3) (cid:3) ln( t ) − − π − ζ (3) + 12415 π + 1283 ln (2)[ π − ln (2)] − (1 / o + y t t n − ( t ) + [189 + 81ln( H ) + 27ln( G )]ln ( t )2+[ − − π − ζ (3) − H ) − G )]ln( t )+ 1576716 + 252 π − π − ζ (3) − (2)[ π − ln (2)]+192Li (1 /
2) + 9ln( H ) + 27ln( G ) o , (A.30)or, numerically, V ≈ g t n − ( t ) + 868ln ( t ) − .
20 ln( t ) + 1957 . o + g y t t n ( t ) − ( t ) + 1220 . t ) − . o + y t t n − . ( t ) + [189 + 81ln( H ) + 27ln( G )]ln ( t )+[ − . − H ) − G )]ln( t ) + 504 .
51 + 9ln( H ) + 27ln( G ) o . (A.31)For the check of renormalization group invariance mentioned at the end of section IV,the scalar field anomalous dimension and beta functions are γ φ = − Q d ln φdQ = ∞ X ℓ =1 π ) ℓ γ ( ℓ ) φ , (A.32) β X = Q dXdQ = ∞ X ℓ =1 π ) ℓ β ( ℓ ) X , (A.33)with the 1-loop contributions: γ (1) φ = 3 y t − g / − g ′ / , (A.34) β (1) λ = − y t + 12 λy t + 24 λ − λg − λg ′ + 9 g / g g ′ / g ′ / , (A.35) β (1) m = m [6 y t + 12 λ − g / − g ′ / , (A.36) β (1) y t = y t (cid:2) y t / − g − g / − g ′ / (cid:3) , (A.37) β (1) g = − g , (A.38) β (1) g = − g / , (A.39) β (1) g ′ = 41 g ′ / . (A.40)The necessary 2-loop contributions are [50–53], [29]: γ (2) φ = 20 g y t − y t / y t g / y t g ′ /
24 + 6 λ − g / g g ′ /
16 + 431 g ′ / , (A.41)3 β (2) λ = − g y t + 30 y t − y t g ′ / − λy t + 80 g y t λ − y t λ + 45 y t λg / y t λg ′ / − y t g / y t g g ′ / − y t g ′ / − λ + 108 λ g +36 λ g ′ − λg / λg g ′ / λg ′ /
24 + 305 g / − g g ′ / − g g ′ / − g ′ / , (A.42) β (2) m = m [40 g y t − y t / − λy t + 45 y t g / y t g ′ / − λ +72 λg + 24 λg ′ − g /
16 + 15 g g ′ / g ′ / , (A.43) β (2) y t = y t (cid:2) − g + 36 g y t − y t + . . . (cid:3) . (A.44)Partial 3-loop contributions are needed here only for the beta function for λ [54–56]: β (3) λ = g y t [64 ζ (3) − /
3] + g y t [480 ζ (3) −
76] + y t [ − ζ (3) − /
4] + . . . . (A.45)Finally, the beta function contributions for the field-independent vacuum energy density are: β (1)Λ = 2( m ) , (A.46) β (2)Λ = (12 g + 4 g ′ − y t )( m ) . (A.47) Acknowledgments:
I thank Hiren Patel for discussions. This work was supported in partby the National Science Foundation grant number PHY-1068369. [1] G. Aad et al. [ATLAS Collaboration], “Observation of a new particle in the search for theStandard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B , 1(2012) [1207.7214],[2] S. Chatrchyan et al. [CMS Collaboration], “Observation of a new boson at a mass of 125 GeVwith the CMS experiment at the LHC,” Phys. Lett. B , 30 (2012) [1207.7235].[3] [ATLAS Collaboration], “Combined measurements of the mass and signal strength of theHiggs-like boson with the ATLAS detector using up to 25 fb − of proton-proton collisiondata,” ATLAS-CONF-2013-014, March 6, 2013.[4] [CMS Collaboration], “Combination of standard model Higgs boson searches and measure-ments of the properties of the new boson with a mass near 125 GeV” CMS-PAS-HIG-12-045,November 16, 2012.[5] S. R. Coleman and E. J. Weinberg, “Radiative Corrections as the Origin of SpontaneousSymmetry Breaking,” Phys. Rev. D , 1888 (1973).[6] R. Jackiw, “Functional evaluation of the effective potential,” Phys. Rev. D , 1686 (1974).[7] M. Sher, “Electroweak Higgs Potentials and Vacuum Stability,” Phys. Rept. , 273 (1989),and references therein.[8] M. Lindner, M. Sher and H. W. Zaglauer, “Probing Vacuum Stability Bounds at the FermilabCollider,” Phys. Lett. B , 139 (1989).[9] P. B. Arnold and S. Vokos, “Instability of hot electroweak theory: bounds on m(H) and M(t),”Phys. Rev. D , 3620 (1991).[10] C. Ford, D. R. T. Jones, P. W. Stephenson and M. B. Einhorn, “The Effective potential and the renormalization group,” Nucl. Phys. B , 17 (1993) [hep-lat/9210033].[11] J. A. Casas, J. R. Espinosa and M. Quir´os, “Improved Higgs mass stability bound inthe standard model and implications for supersymmetry,” Phys. Lett. B , 171 (1995)[hep-ph/9409458].[12] J. R. Espinosa and M. Quiros, “Improved metastability bounds on the standard model Higgsmass,” Phys. Lett. B , 257 (1995) [hep-ph/9504241].[13] J. A. Casas, J. R. Espinosa and M. Quiros, “Standard model stability bounds for new physicswithin LHC reach,” Phys. Lett. B (1996) 374 [hep-ph/9603227].[14] G. Isidori, G. Ridolfi and A. Strumia, “On the metastability of the standard model vacuum,”Nucl. Phys. B , 387 (2001) [hep-ph/0104016].[15] J. R. Espinosa, G. F. Giudice and A. Riotto, “Cosmological implications of the Higgs massmeasurement,” JCAP , 002 (2008) [0710.2484].[16] N. Arkani-Hamed, S. Dubovsky, L. Senatore and G. Villadoro, “(No) Eternal Inflation andPrecision Higgs Physics,” JHEP , 075 (2008) [0801.2399].[17] F. Bezrukov and M. Shaposhnikov, “Standard Model Higgs boson mass from inflation: Twoloop analysis,” JHEP , 089 (2009) [0904.1537].[18] J. Ellis, J. R. Espinosa, G. F. Giudice, A. Hoecker and A. Riotto, “The Probable Fate of theStandard Model,” Phys. Lett. B , 369 (2009) [0906.0954].[19] J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori, A. Riotto and A. Strumia, “Higgsmass implications on the stability of the electroweak vacuum,” Phys. Lett. B , 222 (2012)[1112.3022].[20] S. Alekhin, A. Djouadi and S. Moch, “The top quark and Higgs boson masses and the stabilityof the electroweak vacuum,” Phys. Lett. B , 214 (2012) [1207.0980].[21] F. Bezrukov, M. Y. Kalmykov, B. A. Kniehl and M. Shaposhnikov, “Higgs Boson Mass andNew Physics,” JHEP , 140 (2012) [1205.2893].[22] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori and A. Strumia,“Higgs mass and vacuum stability in the Standard Model at NNLO,” JHEP , 098 (2012)[1205.6497].[23] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio and A. Strumia,“Investigating the near-criticality of the Higgs boson,” [1307.3536].[24] H. Yamagishi, “Coupling Constant Flows and Dynamical Symmetry Breaking,” Phys. Rev.D , 1880 (1981). H. Yamagishi, “Renormalization Group Analysis of Supersymmetric MassHierarchies,” Nucl. Phys. B , 508 (1983).[25] M. B. Einhorn and D. R. T. Jones, “Scale Fixing by Dimensional Transmutation: Supersym-metric Unified Models and the Renormalization Group,” Nucl. Phys. B , 29 (1983).[26] B. M. Kastening, “Renormalization group improvement of the effective potential in massivephi**4 theory,” Phys. Lett. B , 287 (1992).[27] M. Bando, T. Kugo, N. Maekawa and H. Nakano, “Improving the effective potential,” Phys.Lett. B , 83 (1993) [hep-ph/9210228]. “Improving the effective potential: Multimass scalecase,” Prog. Theor. Phys. , 405 (1993) [hep-ph/9210229].[28] M. B. Einhorn and D. R. T. Jones, “The Effective potential, the renormalisation group andvacuum stability,” JHEP , 051 (2007) [hep-ph/0702295].[29] C. Ford, I. Jack and D.R.T. Jones, “The Standard model effective potential at two loops,”Nucl. Phys. B , 373 (1992) [Erratum-ibid. B , 551 (1997)] [hep-ph/0111190].[30] S.P. Martin, “Two loop effective potential for a general renormalizable theory and softlybroken supersymmetry,” Phys. Rev. D , 116003 (2002) [hep-ph/0111209].[31] S.P. Martin, “Three-loop Standard Model effective potential at leading order in strong andtop Yukawa couplings,” Phys. Rev. D , 013003 (2014) [arXiv:1310.7553 [hep-ph]].[32] L. Dolan and R. Jackiw, “Gauge Invariant Signal for Gauge Symmetry Breaking,” Phys. Rev.D , 2904 (1974).[33] J. S. Kang, “Gauge Invariance of the Scalar-Vector Mass Ratio in the Coleman-WeinbergModel,” Phys. Rev. D , 3455 (1974).[34] W. Fischler and R. Brout, “Gauge Invariance in Spontaneously Broken Symmetry,” Phys.Rev. D , 905 (1975). [35] J. -M. Frere and P. Nicoletopoulos, “Gauge Invariant Content of the Effective Potential,”Phys. Rev. D , 2332 (1975).[36] N. K. Nielsen, “On the Gauge Dependence of Spontaneous Symmetry Breaking in GaugeTheories,” Nucl. Phys. B , 173 (1975).[37] R. Fukuda and T. Kugo, “Gauge Invariance in the Effective Action and Potential,” Phys. Rev.D , 3469 (1976).[38] I. J. R. Aitchison and C. M. Fraser, “Gauge Invariance and the Effective Potential,” AnnalsPhys. , 1 (1984).[39] R. Kobes, G. Kunstatter and A. Rebhan, “Gauge dependence identities and their applicationat finite temperature,” Nucl. Phys. B , 1 (1991).[40] D. Metaxas and E. J. Weinberg, “Gauge independence of the bubble nucleation rate in theorieswith radiative symmetry breaking,” Phys. Rev. D , 836 (1996) [hep-ph/9507381].[41] P. Gambino and P. A. Grassi, “The Nielsen identities of the SM and the definition of mass,”Phys. Rev. D , 076002 (2000) [hep-ph/9907254].[42] L. P. Alexander and A. Pilaftsis, “The One-Loop Effective Potential in Non-Linear Gauges,”J. Phys. G , 045006 (2009) [arXiv:0809.1580 [hep-ph]].[43] W. Loinaz and R. S. Willey, “Gauge dependence of lower bounds on the Higgs massderived from electroweak vacuum stability constraints,” Phys. Rev. D , 7416 (1997)[hep-ph/9702321].[44] O. M. Del Cima, D. H. T. Franco and O. Piguet, “Gauge independence of the effective potentialrevisited,” Nucl. Phys. B , 813 (1999) [hep-th/9902084].[45] H. H. Patel and M. J. Ramsey-Musolf, “Baryon Washout, Electroweak Phase Transition, andPerturbation Theory,” JHEP , 029 (2011) [arXiv:1101.4665 [hep-ph]].[46] L. Di Luzio and L. Mihaila, “On the gauge dependence of the Standard Model vacuum insta-bility scale,” arXiv:1404.7450 [hep-ph].[47] N. K. Nielsen, “Removing the gauge parameter dependence of the effective potential by a fieldredefinition,” arXiv:1406.0788 [hep-ph].[48] E. J. Weinberg and A. -q. Wu, “Understanding Complex Perturbative Effective Potentials,”Phys. Rev. D , 2474 (1987).[49] S. P. Martin and D. G. Robertson, “TSIL: A Program for the calculation of two-loop self-energy integrals,” Comput. Phys. Commun. , 133 (2006) [hep-ph/0501132].[50] M. E. Machacek and M. T. Vaughn, “Two Loop Renormalization Group Equations in aGeneral Quantum Field Theory. 1. Wave Function Renormalization,” Nucl. Phys. B , 83(1983).[51] M. E. Machacek and M. T. Vaughn, “Two Loop Renormalization Group Equations in aGeneral Quantum Field Theory. 2. Yukawa Couplings,” Nucl. Phys. B , 221 (1984).[52] I. Jack and H. Osborn, “General Background Field Calculations With Fermion Fields,” Nucl.Phys. B , 472 (1985).[53] M. E. Machacek and M. T. Vaughn, “Two Loop Renormalization Group Equations in aGeneral Quantum Field Theory. 3. Scalar Quartic Couplings,” Nucl. Phys. B , 70 (1985).[54] K. G. Chetyrkin and M. F. Zoller, “Three-loop β -functions for top-Yukawa and the Higgsself-interaction in the Standard Model,” JHEP , 033 (2012) [1205.2892].[55] K. G. Chetyrkin and M. F. Zoller, “ β -function for the Higgs self-interaction in the StandardModel at three-loop level,” JHEP , 091 (2013) [1303.2890].[56] A. V. Bednyakov, A. F. Pikelner and V. N. Velizhanin, “Higgs self-coupling beta-function inthe Standard Model at three loops,” Nucl. Phys. B , 552 (2013) [1303.4364].[57] S. P. Martin and D. G. Robertson, to appear.[58] S. P. Martin, “Two loop effective potential for the minimal supersymmetric standard model,”Phys. Rev. D , 096001 (2002) [hep-ph/0206136].[59] S. P. Martin, “Complete two loop effective potential approximation to the lightest Higgs scalarboson mass in supersymmetry,” Phys. Rev. D67