Three-loop QCD corrections to the electroweak boson masses

I find the three-loop corrections at leading order in QCD to the physical masses of the Higgs, W, and Z bosons in the Standard Model. The results are obtained as functions of the $\overline{\rm{MS}}$ Lagrangian parameters only, using the tadpole-free scheme for the vacuum expectation value. The dependences of the computed masses on the renormalization scale are found to be smaller than present experimental uncertainties in each case. In the case of the Higgs boson mass, the new result is the state-of-the-art, while the results for $W$ and $Z$ are in good numerical agreement with corresponding results in the on-shell and hybrid schemes. These results are now included in the SMDR (Standard Model in Dimensional Regularization) computer code.

Since the discovery of the Higgs boson in 2012, the Standard Model is a mathematically complete theory, for which precision calculations can be performed. In addition to providing a test of the agreement of the theory with experiment, this allows us to obtain accurate results for the shortdistance Lagrangian parameters, suitable for matching to candidate ultraviolet completions. The goal of this paper is to report the 3-loop QCD contributions to the pole masses of the W , Z, and Higgs bosons in the Standard Model. In the case of the Higgs boson mass, the result obtained is the new state-of-the-art result, including the complete set of 2-loop effects as well as the three-loop terms proportional to α 2 S y 2 t including all momentum-dependent effects, as well as the three-loop terms proportional to α S y 4 t and y 6 t in the approximation that M 2 h ≪ M 2 t .
The results below are given in the pure MS renormalization scheme [1,2] based on dimensional regularization [3][4][5][6][7], so that all independent inputs are running Lagrangian parameters. The calculation is also based on the tadpole-free scheme for the Higgs vacuum expectation value (VEV), which is defined to be the minimum of the exact Landau gauge effective potential, currently known in an approximation at full 3-loop order [8][9][10][11] with the leading 4-loop order QCD part [12] and resummation of the Goldstone boson contributions [13,14]. The tadpole-free VEV scheme has a formally faster convergence in perturbation theory than schemes based on a tree-level VEV definition, since in the latter the tadpole diagrams necessarily introduce inverse powers of the Higgs self-coupling λ. The price to be paid for this improvement is that the validity of the calculations is restricted to the Landau gauge fixing prescription in the electroweak sector.
Previous 2-loop calculations of the W and Z masses have been given in refs. [15][16][17][18], using the tree-level definition for the VEV. In addition, there is a long history of calculations of the ρ parameter including up to 4-loop order QCD contributions , which can be used to relate the W boson on-shell mass to the Z-boson mass. The present paper relies on a quite different organization of perturbation theory, by taking all physical masses as outputs including the W and Z boson pole masses separately, rather than using the Z boson on-shell mass as an input. The complete 2-loop W and Z boson pole squared masses in the scheme adopted in this paper were given in refs. [42] and [43] respectively. The present paper will add the 3-loop QCD contributions to those results in a consistent way.
In the case of the Higgs boson pole squared mass, ref. [44] provided the mixed QCD/electroweak parts, ref. [45] gave results in the gauge-less limit in which g, g ′ are neglected in the 2-loop part, and ref. [46] gave an interpolating formula for the full 2-loop approximation in a hybrid MS /on-shell scheme. In ref. [47], the full 2-loop corrections were extended to include the 3-loop contributions in the gauge-less effective potential limit (formally, g 2 3 , y 2 t ≫ λ, g 2 , g ′2 , where g 3 , g, g ′ are the gauge couplings, y t is the top-quark Yukawa coupling, and λ is the Higgs self-coupling) using the pure MS tadpole-free scheme. The present paper will extend this further to include the momentumdependent parts of the leading QCD contribution to the Higgs boson self-energy in the calculation of the pole squared mass.
To specify notation, the complex pole squared masses for the electroweak bosons are each given in the loop-expansion form with X = W , Z, and h. The complete 2-loop contributions given in refs. [42,43,47] were written in terms of master integrals defined in refs. [48,49], the latter of which provided a computer program TSIL (Two-loop Self-energy Integral Library) for their efficient numerical evaluation. The computer program SMDR (Standard Model in Dimensional Regularization) [50] incorporates these calculations of the W, Z, and Higgs physical masses and many other results within the pure MS tadpole-free scheme, matching observables to Lagrangian parameters. Another public code mr [51] provides similar functionality, but using the tree-level VEV scheme. For the vector bosons, it is important to note that the standard practice in experimental papers and by the Review of Particle Properties (RPP) [52] from the Particle Data Group (PDG) is to report the on-shell masses found from a variable-width Breit-Wigner linewidth fit, which should be related to the complex pole mass and width M X and Γ X defined in eq. (1.1) by (1.4) (In this paper, the superscript "PDG" refers to the convention used by the PDG, and not to the averaged experimental results produced by the PDG in the RPP.) To add to the potential for confusion, in refs. [42,43,47] by the present author, and many publications by other authors, a different parameterization for complex pole masses has been used, denoted here by: which is related to the M and Γ in eq. (1.1) by The (M PDG , Γ PDG ) and (M ′ , Γ ′ ) parameterizations can be considered to contain the same information as (M, Γ), through the defining relations in eqs. (1.2)-(1.4) and (1.6)-(1.7). However, as emphasized in a recent paper [53], the (M, Γ) parameterization defined by eq. (1.1) has the clear advantage that Γ = 1/τ is precisely the inverse mean lifetime of the particle, unlike Γ PDG and Γ ′ . In the following s pole will be computed, but the information that it contains must be converted to M PDG to compare directly with the results quoted by the PDG and experimental collaborations. The 3-loop integrals to be used below have been defined and discussed in sections IV, VI, and VII of ref. [55]. The master integrals are given there as a renormalized ǫ-finite basis, defined so that expansions of integrals to positive powers in ǫ will never be needed, even when the results of the present paper are (eventually) extended to 4-loop order or beyond. Denoting the lists of 1-loop, 2-loop, and 3-loop renormalized ǫ-finite master integrals by I j , and I (3) j , respectively, then † After the first version of the present paper, the CDF collaboration released [54]  the general form of a 3-loop contribution to the pole mass of X = W, Z, or h is where all of the coefficients c (3) , c (2,1) j,k , . . . , c (0) are dimensionless MS couplings multiplied by rational functions of the MS top-quark squared mass and either s = W, Z, or h as appropriate, where The VEV v is defined to be the minimum of the MS effective potential in Landau gauge at all orders in perturbation theory, so that the sum of all Higgs tadpole diagrams vanishes. Note that the name of each particle is being used as a synonym for the tree-level MS squared mass in the tadpole-free scheme. (All other fermions are taken to be massless, except in the 1-loop parts ∆ (1) X .) Note also that the tree-level MS squared masses t, W , Z, and h are not gauge invariant, but are specific to Landau gauge, due to their dependence on the VEV. However, as is well-known, the complex pole masses [and thus the PDG-convention masses for W and Z, defined by eqs. (1.2)-(1.4)] are gauge-invariant.
The loop integrals include logarithmic dependences on the MS renormalization scale Q, written in this paper in terms of for the external momentum invariant s, which has a positive infinitesimal imaginary part. In the 3-loop parts ∆ X , the integrals will always be evaluated at external momentum invariant equal to the tree-level squared mass, s = W , Z, or h. This is just as consistent as choosing to evaluate them at the (real part of) the corresponding pole squared mass instead, as the difference is of 4-loop order, and numerically small.
In order to provide more opportunities for checks, the results below will be given in terms of SU (3) c group theory quantities ( 1.15) Here N c is the number of colors, C G and C F are the quadratic Casimir invariants of the adjoint and fundamental representations respectively, T F is the Dynkin index of the fundamental representation, and n g is the number of fermion generations. For numerical results shown below, I will use a benchmark Standard Model designed to give output parameters in agreement with the current central values of the 2021 update of the 2020 RPP [52]: Using the latest version 1.2 of the computer program SMDR [50], which incorporates the new results of the present paper, these are best fit by the MS input parameters (using the tadpole-free scheme for the Landau-gauge VEV, and writing g 3 for the QCD coupling in the full 6-quark Standard Model theory): Here I have included many more significant digits than justified by the theoretical errors, merely for the sake of reproducibility. These MS quantities can be run to a different renormalization scale choice Q, where the pole squared masses can be recomputed. In the idealized case, the pole squared masses, being observables, would be independent of the scale Q at which they are computed. The renormalization group running is carried out using the state-of-the-art beta functions for the Standard Model. The 2-loop and 3-loop beta functions were found in [56][57][58][59][60] and [61][62][63][64][65][66][67][68][69], respectively. The 4-loop beta function for the QCD coupling g 3 was found in [70][71][72][73][74] in the approximation that only g 3 , y t , and λ are included. The pure QCD 5-loop beta functions were obtained in [75,76], and the 4-loop and 5-loop QCD contributions to the quark Yukawa beta functions were obtained in refs. [77,78] and ref. [79] respectively, and the 4-loop QCD contributions to the beta function of the Higgs self-coupling λ were obtained from [12,80]. Finally, the complete 4-loop beta functions for the three gauge couplings have been provided by [81]. All of these results have been included in the latest version of the code SMDR, which was used to carry out the numerical computations described below. The code also implement results for multi-loop threshold matching 1: Three-loop contribution to the Z boson mass from diagrams involving two triangle quark loops, which give a non-vanishing contribution with a consistent treatment of the axial vector coupling. These contributions are individually divergent for each of (q, , but are finite and gauge invariant after the combination. Contributions involving sums over other (q, q ′ ) quark doublet combinations vanish in the massless quark limit.

II. THE Z BOSON POLE MASS
Consider the Z-boson complex pole squared mass, s Z pole , in the form of eq. (1.1). The complete 1-loop and 2-loop contributions ∆ (1) Z and ∆ (2) Z were given in the tadpole-free pure MS scheme in ref. [43]. The 3-loop QCD part can be split into contributions from 13 distinct classes of self-energy diagrams with different group theory structures, using the quantities defined in eq. (1.15): where the tree-level couplings of the Z boson to up-type and down-type quarks are Most of the three-loop diagrams are straightforward to set up, and can be carried out with a naive treatment of γ 5 , taken to anti-commute with all of the other gamma matrices. The known exception to this is the double triangle diagrams shown in Figure 2.1, which feature two distinct triangle quark loops each containing a γ 5 from the axial vector coupling to the Z boson. (The vector couplings to the Z boson give vanishing contributions for the sum of these diagrams.) The contributions from (q, q ′ ) = (t, t), (t, b), (b, t), (b, b) are separately divergent, but their sum is finite and gauge invariant. Therefore, for these diagrams only, one can use the prescription [104,105] based on the 't Hooft-Veltman treatment [6] of γ 5 , and then carry out the Lorentz algebra in 4 dimensions before reducing to master integrals in d dimensions. The result is the contribution ∆ in eq. (2.1). The contributions from diagrams with one or both of q and q ′ summed over the other quark doublets (u, d) and (c, s) vanish, because the axial couplings a q L − a q R for down-type and up-type quarks have the same magnitude and opposite sign, and they are being treated as mass degenerate (specifically, massless). The result for general non-zero s = Z found here reduces to ∆ (3,i) Z → 21ζ 3 for s = 0, which agrees with the original calculation in that limit [106] and with the corresponding contribution to the ρ parameter obtained in [27,28,39].
The contributions from the diagrams in which the Z boson couples directly to a single massless (in the present approximation, non-top) quark loop are relatively simple, and can be written as: contains a top-quark loop that corrects a gluon propagator, rather than connecting to the external Z boson. The remaining contributions in eq. (2.1) are much more complicated, and are given in an ancillary file DeltaZ3 provided with this paper. Each of the contributions has the form of eq. (1.8), with master integrals chosen in ref. [55]: 10) , H(0, t, t, t, 0, t), I 4 (t, t, t, t), I 5a (t, 0, t, 0, t), I 5b (0, t, t, t, t), I 5c (t, t, t, t, t), I 6c (t, t, t, 0, t, t), I 6c2 (t, t, t, 0, 0, 0), I 6d (0, t, t, t, t, 0), I 6d (t, 0, t, 0, t, 0), I 6d (t, 0, t, t, 0, t), I 6e (0, 0, 0, 0, t, t), I 6e (0, t, t, t, 0, t), I 6e (t, t, t, 0, t, t), I 6f (0, 0, 0, 0, t, t), However, in eq. (2.7) above, I have chosen to write the expression for ∆ (3,l) Z in terms of candidate master integrals that were solved for in ref. [55], rather than the master integrals listed above (which are a subset of the ones listed in eq. (7.4) in ref. [55], joined by B(0, 0) and ζ 3 and ζ 5 from the integrals with all propagators massless). This simplifies the expression somewhat, because the integrals used in eq. (2.7) have the same propagator structures as descendants of the underlying Feynman diagrams for the ∆ (3,l) Z contribution. As a check of eq. (2.1), I have verified that the full expression for the observable s Z pole is renormalization group invariant through 3-loop terms proportional to g 4 3 , using the derivatives of the master integrals with respect to Q found in the ancillary file QddQ of ref. [55].
For practical numerical evaluation, after using the Standard Model group theory values in eq. (1.15), and applying the expansions for the master integrals in the ancillary file Ievenseries of ref. [55], I find: where the series expansions of δ Z 1 , δ Z 2 , δ Z 3 , and δ Z 4 are given in the ancillary file DeltaZ3series to order r 18 Z , where (2.13) The contribution δ Z 1 isolates the results form the double triangle diagrams in Figure 2.1. The series expansion coefficients are given both numerically, and analytically in terms of L t , L −Z , and the constants ζ 3 , ζ 5 , and 14) The series converge for all r Z < 1, which is clearly satisfied in actuality. The first few terms in the expansions are It is interesting to note that in the expansion in small r Z , the sub-leading contribution is numerically comparable to (or even larger than, for smaller Q) the leading contribution obtained by r Z = 0. This is due mostly to the term proportional to r Z L 2 −Z in the contribution eq. (2.18) from massless quark loops, because of the large magnitude of the coefficient −122.667 and because L 2 −Z = [−iπ + ln(Z/Q 2 )] 2 provides up to an order of magnitude enhancement. The resulting contribution of eq. (2.12) has now been included in the latest version 1.2 of the code SMDR [50]. GeV. To obtain the results in the figure, the MS input parameters are run to other MS scales Q using the most complete available renormalization group equations (as listed at the end of the Introduction), and s Z pole is then re-calculated. In the idealized case, the results should not depend on Q. I find that with inclusion of the 3-loop QCD corrections, the scale dependence of M Z is remarkably small, less than 0.8 MeV as Q is varied between 50 GeV and 220 GeV. However, given the larger scale dependence found in section IV for the similar case of the W boson mass, I surmise that this very mild scale dependence is partly accidental, and the actual theoretical error due to neglecting higher order contributions is likely to be larger.
The scale dependence of Γ Z shown in the right panel of Figure 2.2 is less mild, and not so much improved over the complete 2-loop result, as it varies by a total of about 4 MeV (between minimum and maximum) as Q is varied between 80 GeV and 220 GeV. Note that this determination of Γ Z from the complex pole mass (in which the leading contribution arises only as a 1-loop effect) is essentially one loop order less accurate than a direction calculation of the Z-boson decay width (in which the leading contribution is a tree-level effect).

III. THE HIGGS BOSON POLE MASS
Next, consider the complex pole mass s h pole for the Standard Model Higgs boson, written in the form of eq. (1.1). In this section, I extend the results of ref. [47] to include the momentumdependent 3-loop self-energy corrections to ∆ (3) h that are proportional to g 4 3 y 2 t t. Also included below are the 3-loop contributions proportional to g 2 3 y 4 t t and y 6 t t, in an effective potential approximation, which amounts to g 2 3 , y 2 t ≫ λ, g 2 , g ′2 . For the y 6 t t part, I provide below an improvement over the result in [47]. Together with the full 2-loop results, these constitute the most complete calculation of the Standard Model Higgs boson mass that is presently available.
The functions ∆ (1) h and the QCD part of ∆ (2) h were given in eqs. (2.46) and (2.47) in ref. [47], and are evaluated at s = Re[s h pole ], determined by iteration. The remaining, non-QCD part of ∆ (2) h was given in an ancillary file of ref. [47], where the master integrals were also evaluated at s = Re[s h pole ]. However, in the present paper, I adopt a slightly different organization by evaluating the non-QCD part of ∆ (2) h as exactly the same function but evaluated instead at s = h, which is consistent up to 3-loop terms of order y 6 t t. This allows an easier extension to 3-loop order, as indicated below.
For the leading QCD part of ∆ h proportional to g 4 3 y 2 t t, the new result can be written in terms of the contributions of four distinct classes of self-energy diagrams characterized by their group theory structures: The results for ∆ are somewhat lengthy, and so are given in the ancillary file DeltaH3 provided with this paper. They are written in terms of the same list of 3-loop self-energy master integrals as for the Z boson, listed in eqs. (2.9)-(2.11), with the exceptions that I 8b (t, 0, t, t, t, t, t, 0) is also needed in I (3) , and ζ 5 , I 6e (0, 0, 0, 0, t, t), I 6f (0, 0, 0, 0, t, t), I 6f 5 (0, 0, 0, 0, t, t), I 7e (0, 0, 0, 0, 0, t, t), and I 7e (0, 0, t, t, t, 0, 0) are not needed, and of course one should use s = h rather than s = Z.
Using the expansions of the master integrals given in ref. [55], and setting s = h in ∆ (3),g 4 3 y 2 t t h (which is consistent up to terms of 4-loop order), and plugging in the group theory constants from eq. (1.15), the result becomes a power series in with coefficients that depend on L t ≡ ln(t/Q 2 ) and L −h ≡ ln(h/Q 2 ) − iπ and the constants ζ 3 and c ′ H from eq. (2.14). The expansion converges for r h < 1, and does so rapidly for the value realized in the Standard Model. It is given to order r 24 h in the ancillary file DeltaH3series, both in analytic and numerical forms. The first few terms of the numerical form are: h proportional to g 2 3 y 4 t t, the effective potential approximation gives the second line of eq. (3.3) of ref. [47], which is not improved on in the present paper, but is reproduced here for reference and comparison: It is interesting that ∆ , despite the parametric relative enhancement N c g 2 3 /y 2 t of the former. In the approximation r h = 0, this effect was noted in refs. [10,47] (see the discussion involving eqs. (6.21)-(6.28) of the former reference) as the result of an unexplained but dramatic near-cancellation, and is found here to be not changed by the inclusion of terms higher order in r h .
Finally, for the part of ∆ (3) h proportional to y 6 t t, the effective potential approximation of ref. [47] can be improved on slightly as follows. In the present paper, the non-QCD part of ∆ (2) h is evaluated using master integrals with external momentum invariant h rather than Re[s h pole ]. Then, due to the fortunate circumstance that the leading 1-loop behavior of s h pole − h in the limit y 2 t ≫ λ, g 2 , g ′2 is proportional to L t : we can fully repair the error in the 3-loop part (caused by using h rather than Re[s h pole ] in the 2-loop part), simply by requiring renormalization group invariance of the pole mass. This allows inference of the complete dependence proportional to y 6 t tL t , due to the explicit dependence on Q. By demanding (and checking) renormalization group invariance of s h pole through terms of 3-loop order in the approximation g 2 3 , y 2 t ≫ λ, g 2 , g ′2 , I find that the end result for the leading non-QCD 3-loop contribution is that eq. (3.4) of ref. [47] should be replaced by: where the analytic forms of the decimal coefficients are This result differs from eq. (3.4) of ref. [47] by terms that vanish when L t = 0, consistent with the approximation made in that reference.
To recapitulate: in order to consistently include the 3-loop results given above, the non-QCD part of ∆ (2) h found in the ancillary file of ref. [47] should use s = h in the evaluation of the integrals, while ∆ (1) h and the QCD part of ∆ (2) h provided in that reference should use s = Re[s h pole ] determined by iteration. All of these results for the Higgs boson pole mass have now been implemented in version 1.2 of the computer code SMDR [50]. The new contributions found in this paper give the best approximation available at this writing, but still imply a scale dependence of several tens of MeV. For example, the calculated M h decreases by about 56 MeV when Q is varied from 100 GeV to 200 GeV, for fixed values of the MS input parameters. This provides a lower bound on the theoretical error, and suggests that a still more refined calculation of the Higgs pole mass, to include 3-loop electroweak parts and even leading 4-loop contributions, would be worthwhile, since the experimental uncertainty on M h from future collider experiments may well be smaller [107]. It is also possible [108] to refine further the gaugeless limit by including momentum-dependent parts of the Higgs boson self-energy function.
A famous feature of the observed Higgs boson mass is that the Standard Model with no extensions can then have the Higgs self-coupling λ run negative at a scale that is far above the electroweak scale but below the Planck scale, implying a possibly metastable electroweak vacuum. This is illustrated in Figure 3.2, using the latest experimental values and the results of this paper to relate M h to λ in the most accurate available way. As is well-known (see for example refs. [109] and [44][45][46]), the scale of possible instability is lowered if the top-quark mass is higher, or the QCD coupling is lower, or the Higgs mass is lower, than their benchmark values, while it is possible for the instability to be avoided up to the Planck scale if the deviations are in the opposite directions. While improved formulas and experimental values for M h are welcome, the dominant uncertainty in these instability discussions comes from M t (or y t ), and the second most important uncertainty is that of α W were given in ref. [42]. The 3-loop QCD part splits into 8 distinct contributions with different group theory structures: The four contributions from diagrams in which the W boson couples directly to massless quarks are relatively simple: In fact, ∆ I 7a (0, 0, 0, 0, t, t, t), I 7a (0, 0, t, t, 0, 0, 0), I 7a (0, t, t, 0, 0, t, 0), I 7a (t, t, 0, 0, t, t, 0), I 7a5 (t, t, 0, 0, t, t, 0), I 7b (0, 0, t, 0, t, 0, 0), I 7c (0, 0, t, t, 0, 0, 0), I 7d (0, t, 0, t, t, 0, t), I 7e (0, t, t, 0, 0, 0, 0), I 8b (0, 0, 0, t, t, 0, 0, t), I 8c (0, 0, 0, t, t, 0, 0, t), I pk 8c (t, t, t, 0, 0, 0, 0, 0)}. (4.8) I have checked that eq. (4.1) gives a pole mass s W pole that is renormalization group invariant through 3-loop terms of order g 4 3 , using the derivatives of the master integrals with respect to Q found in the ancillary file QddQ of ref. [55].
For practical numerical evaluation, after plugging in the Standard Model group theory values in eq. (1.15), and applying the expansions for the master integrals in the ref. [55] ancillary files Ioddseries and Ievenseries (the latter being needed only for the contribution ∆  and the coefficients involve L t = ln(t/Q 2 ) and L −W = ln(W/Q 2 ) − iπ = 2 − B(0, 0) s=W +iǫ , as well as ζ 2 , ζ 3 , ζ 4 , ζ 5 , c ′ H from eq. (2.14), and Note that δ W 2 is the same as δ Z 4 appearing in eqs. (2.12) (2.18) with the replacement r Z → r W . The series for δ W 1 and δ W 2 converge for ρ W < 1 and r W < 1, respectively, which is clearly satisfied by the relevant value of W/t in the Standard Model.
The numerical form of the first few terms in the series are As in the case of the Z boson, it is interesting to note that in this expansion in small W/t, the subleading contribution is numerically comparable to or larger than the leading contribution (obtained by ρ W = r W = 0), depending on the choice of Q. This is due mostly to the term proportional to r W L 2 −W in the contribution from massless quark loops, because of the large magnitude of the coefficient −122.667 and because L 2 −W = [−iπ + ln(W/Q 2 )] 2 provides up to an order of magnitude enhancement.
The contribution ∆  Figure 2.2, but compares quite favorably to the present experimental uncertainty of 12 MeV. The range for M PDG W from the average of experimental data released through 2021 is 80.379 ± 0.012 GeV. The CDF collaboration has recently produced a result that is substantially higher, 80.4335 ± 0.0064 stat ± 0.0069 syst GeV, which is in stark disagreement with the Standard Model prediction. These results are also shown in Figure 4.1. As seen in the right panel of Figure 4.1, the total variation in Γ W as Q varies from 60 GeV to 220 GeV is about 3.5 MeV, but the spread is only about 2.3 MeV as Q varies from 80 GeV to 180 GeV. These scale variations are improved over the full 2-loop order calculation found in ref. [42]. For comparison, the largest parametric uncertainty contributing to the M W prediction is that of the top-quark pole mass M t . If one fixes eqs. (1.16) and (1.17) as a reference model, and then adjusts the Standard Model inputs to fit varying M t , M Z , ∆α (5) had , and α from SMDR v1.2 (incorporating the results of this paper) in the pure MS scheme to the corresponding results in the on-shell scheme using the interpolation formula in ref. [38], and to those in the hybrid MS-on-shell scheme of ref. [17], as a function of the top-quark pole mass. The other on-shell parameters M PDG Z , G µ , α S (M Z ), ∆α (5) had , and M h are chosen to be the same, and equal to the data given in from eq. (1.16) from the 2021 update to the 2020 RPP, so that the results are directly comparable. (In the MS scheme, this entails doing a fit to determine the Lagrangian parameters, which is readily accomplished using the C function SMDR Fit Inputs or the interactive command-line tool calc fit -int.) The pure MS scheme gives results between those of the on-shell and hybrid schemes, with a total spread between the three schemes of about 4.5 MeV.
S (M Z ), ∆α (5) had , and M h from eq. (1.16). The solid black line is the pure MS scheme result, obtained using SMDR v1.2 incorporating the results of this paper. The short dashed (blue) line is the on-shell scheme result, obtained from the interpolating formula in ref. [38]. The long dashed (red) line is the result from the hybrid MS-on-shell scheme of ref. [17]. Also shown are the experimental central values and 1σ ranges for M PDF W as given by the 2021 update to the 2020 RPP, and from the 2022 result from CDF [54].

V. OUTLOOK
In this paper, I have reported the 3-loop QCD contributions to the W , Z, and Higgs boson physical masses in the Standard Model, in the pure MS renormalization scheme with a tadpole-free treatment of the Higgs VEV. The results show improved renormalization group scale independence, especially for the W and Z boson cases, and in all three cases the scale variation is less than the present experimental uncertainty. Alternative methods based on on-shell type schemes have already included 4-loop QCD contributions through the rho parameter, but it is not clear that these should be numerically more important than 3-loop mixed and pure electroweak contributions. The results of this paper have all been incorporated in the latest version 1.2 of the code SMDR [50]. Further improvements in the approach of the present paper could come from computing all of the remaining 3-loop self-energy contributions to the pole masses, which in the case of the most general diagrams will be a challenging but perhaps not insurmountable goal.