boson pole mass at twoloop order in the pure scheme
Abstract
I obtain the complex pole squared mass of the boson at full twoloop order in the Standard Model in the pure renormalization scheme. The input parameters are the running gauge couplings, the topquark Yukawa coupling, the Higgs selfcoupling, and the vacuum expectation value that minimizes the Landau gauge effective potential. The effects of nonzero Goldstone boson mass are resummed. Within a reasonable range of renormalization scale choices, the scale dependence of the computed pole mass is found to be comparable to the current experimental uncertainty, but the true theoretical error is likely somewhat larger.
Contents
I Introduction
One of the cornerstone physical observables of the Standard Model is the boson mass. The experimental value that is usually quoted is obtained using a BreitWigner parametrization with a variable width, and is given in ref. RPP from a fit to LEP data as:
(1.1) 
This is related Bardin:1988xt ; Willenbrock:1991hu ; Sirlin:1991fd to the real part of the complex pole squared mass (with a constant width) according to:
(1.2)  
(1.3) 
In general, the complex pole squared mass is a physical observable Tarrach:1980up ; Stuart:1991xk ; Stuart:1992jf ; Passera:1996nk ; Kronfeld:1998di ; Gambino:1999ai , independent of the choice of renormalization scheme and scale and the choice of gauge fixing.
In this paper, I report a calculation, at full 2loop order, of the complex pole squared mass parameters and , using the pure scheme. The input parameters in this scheme are the running renormalized quantities
(1.4) 
where the first three are the Standard Model gauge couplings, is the topquark Yukawa coupling, is the Higgs selfcoupling, and the vacuum expectation value (VEV) is defined here to be the minimum of the full radiatively corrected effective potential in the Landau gauge. The normalizations used here for and are fixed by writing the treelevel Higgs potential as
(1.5) 
where the canonically normalized doublet Higgs field has VEV , and is a negative Higgs squared mass parameter. The minimization condition that relates to (allowing the latter to be eliminated) is presently known at full 2loop order FJJ augmented by all 3loop contributions at leading orders in both and Martin:2013gka . Goldstone boson mass effects are resummed in this relation using Martin:2014bca ; EliasMiro:2014pca ; this effect is usually numerically small but is conceptually important, and in any case leads to simpler formulas.
Other definitions of the Higgs VEV can be found in the literature. One alternative (for example, see refs. Jegerlehner:2001fb ; Jegerlehner:2002em ; Kniehl:2015nwa ) is to instead define the VEV as the minimum of the treelevel potential, so . This has the disadvantage that one must include tadpole diagrams explicitly. Also, one is then expanding around a point that differs from the true radiatively corrected vacuum, so perturbation theory converges less quickly, at least formally and for generic choices of the renormalization scale. Indeed, in the large limit, the loop expansion parameter is rather than the usual . [For more details, see for example refs. Martin:2014bca ; Martin:2015lxa , and the discussion surrounding eq. (2.34) below.] The reason for the in the denominator is that the tadpole diagrams have a Higgs propagator at zero momentum, which is just the reciprocal of the Higgs squared mass.
Another alternative (see for example ref. Degrassi:2014sxa ) is to define the VEV so that the sum of tadpole graphs in Feynman gauge vanishes. However, the Landau gauge effective potential is easier to compute to higher orders, and avoids renormalization of the gaugefixing parameter, making it arguably a more convenient choice as a standard.
The pure scheme is an alternative to onshell and hybrid schemes, which have been used for many precision studies of the mass and the electroweak sector. For a selection of some important related results in that approach, see refs. Sirlin:1980nh ; Marciano:1980pb ; Sirlin:1983ys ; Djouadi:1987gn ; Consoli:1989fg ; Kniehl:1989yc ; Djouadi:1993ss ; Avdeev:1994db ; Chetyrkin:1995ix ; Chetyrkin:1995js ; Degrassi:1996mg ; Degrassi:1996ps ; Degrassi:1997iy ; Passera:1998uj ; Freitas:2000gg ; Awramik:2002wn ; Faisst:2003px ; Awramik:2003ee ; Awramik:2003rn ; Schroder:2005db ; Chetyrkin:2006bj ; Boughezal:2006xk , and for reviews see refs. RPP ; Sirlin:2012mh .
Ii Complex pole mass of the boson at 2loop order
To obtain the boson pole squared mass, one begins with the symmetric matrix of neutral gauge boson transverse selfenergy functions, for :
(2.1) 
where , with the external momentum, using a metric with Euclidean or (,,,) signature. These are obtained by calculating, in the theory in dimensions with bare parameters, the sum of 1particleirreducible 2point Feynman diagrams for , followed by projecting with . The pole squared mass is then the solution of
(2.2) 
Here,
(2.3) 
is the bare, treelevel, squared mass of the boson. Solving eq. (2.2) iteratively, one obtains to 2loop order:
(2.4)  
Instead of computing separate counterterm diagrams, the calculation described here was done in terms of only bare quantities , , , , , , , and then translated to renormalized running quantities , , , , , at the end. Tadpole diagrams need not be calculated, because they automatically sum to zero, due to the defining condition that the VEV is the minimum of the effective potential. Using the minimization condition for the Landau gauge effective potential given in ref. Martin:2014bca , the parameter (and the Goldstone boson squared mass) are eliminated. These procedures are the same as described in refs. Martin:2014cxa ; Martin:2015lxa , and so most details will not be repeated here. An exception is that the 2loop translation of the gauge couplings from bare to renormalized couplings is needed, to go along with eqs. (2.5)(2.24) of ref. Martin:2014cxa and eqs. (2.3)(2.10) of ref. Martin:2015lxa :
(2.5) 
where
(2.6)  
(2.7)  
(2.8) 
and is the regularization scale, related to the renormalization scale by As in refs. Martin:2014cxa ; Martin:2015lxa , the results are reduced, using the Tarasov recurrence relations Tarasov:1997kx to a set of 1loop basis integrals and 2loop basis integrals , following the notations and conventions of refs. Martin:2003qz ; TSIL . The program TSIL TSIL can be used to automatically and efficiently evaluate these basis integrals numerically. Where possible, TSIL takes advantage of analytical results in terms of polylogarithms, which were given in refs. Martin:2003qz ; TSIL ; Broadhurst:1987ei ; Gray:1990yh ; Davydychev:1992mt ; Davydychev:1993pg ; Scharf:1993ds ; Berends:1994ed ; Berends:1997vk . In many cases, analytical results for the basis integrals are not available, so TSIL employs RungeKutta solution of differential equations in the external momentum invariant Martin:2003qz , similar to that suggested in ref. Caffo:1998du .
The final result for the 2loop boson complex pole mass can be written as:
(2.9) 
In the following,
(2.10)  
(2.11)  
(2.12)  
(2.13) 
are the treelevel squared masses of the boson, boson, top quark, and Higgs boson, respectively, and the couplings of the quarks and leptons to the boson are:
(2.14) 
for , where
(2.15)  
(2.16)  
(2.17)  
(2.18)  
(2.19)  
(2.20) 
Also, , and
(2.21) 
are the the numbers of flavors of twocomponent quarks and leptons of each gauge transformation type, and and and and , respectively. The quantities , , , , and are kept general in the following as a way of tagging different fermion contributions, although they are all equal to 3 in the Standard Model.
The 1loop contribution is then:
(2.22)  
where the fermion 1loop integral functions are:
(2.23)  
(2.24)  
(2.25) 
The basis integrals , , , and , and other integral functions below, are always evaluated at the external momentum invariant and renormalization scale . The bottomquark, taulepton, and other fermion masses have been neglected for simplicity, because even at 1loop order they make a difference of less than 1 MeV in the real pole mass. However, they can easily be restored in the 1loop part by following the example of the topquark terms in the obvious way.
The 2loop QCD contribution can also be written in terms of the basis integral functions in a few lines:
(2.26)  
where:
(2.27)  
(2.28)  
(2.29)  
The 2loop nonQCD contribution to the boson pole squared mass has the form:
(2.30) 
where the list of 1loop basis integrals is
(2.31) 
and the list of necessary 2loop basis integrals is:
(2.32)  
The coefficients and and and are quite lengthy, so they will not be listed in print here. Instead, they are listed in electronic form in an ancillary file provided with the arXiv source for this article, called coefficients.txt. They are ratios of polynomials of , , , , and . As usual, these coefficients are not unique, because of special identities that relate different basis integrals in cases where the masses are not generic.
For each of the fivepropagator integrals for which analytical
results are not available, the main TSIL RungeKutta evaluation function
TSIL_Evaluate
simultaneously computes all of the subordinate
integrals , , obtained by removing one or more propagator
lines. Therefore, only 11 calls of TSIL_Evaluate
are required (in
addition to the relatively fast evaluation of the integrals that are
known in terms of polylogarithms), and in total the numerical
computation takes well under 1 second on modern computer hardware.
I performed a number of stringent analytical checks on the calculation, similar to those described for the calculations of the Higgs and boson pole masses in Martin:2014cxa ; Martin:2015lxa . First, is free of poles in . The cancellation of these poles relies on agreement between the divergent parts of the loop integrals performed here and the counterterm coefficients which can be obtained from the 2loop scalar anomalous dimension and functions from refs. MVI ; MVII ; Jack:1984vj ; MVIII . Second, poles and logs of the Goldstone boson squared mass were checked to cancel after the resummation described in Martin:2014bca ; EliasMiro:2014pca . Third, I checked the cancellations between contributions from unphysical vector propagator components with poles at 0 squared mass and the corresponding Landau gauge Goldstone boson propagators. This ensures the absence of unphysical imaginary parts of the complex pole squared mass. Note that in the case . Next, I checked the absence of singularities in various formal limits (none of which are close to being realized in the actual parameters of the Standard Model), in which one or more of the following quantities vanish: , , , , , , , and . This is despite the fact that many of the individual 2loop coefficients do have singularities in one or more of those cases; nontrivial relations between basis integrals are responsible for the smooth limits of the total. Finally, I checked analytically that the complex pole squared mass is renormalization group scaleinvariant up to and including all terms of 2loop order, using
(2.33) 
where is the Higgs anomalous dimension, and . In the conventions used here, the derivatives of the 1loop basis integrals with respect to squared masses are listed in eqs. (A.5) and (A.6) of ref. Martin:2014cxa , while the derivatives of the 1loop and 2loop basis integrals with respect to the renormalization scale can be found in eqs. (4.7)(4.13) of ref. Martin:2003qz . The beta functions and scalar anomalous dimension are listed in refs. MVI ; MVII ; Jack:1984vj ; MVIII ; FJJ . In the next section, a numerical check of the invariance will be shown.
In refs. Jegerlehner:2001fb ; Jegerlehner:2002em , a calculation of the boson pole mass in the pure scheme has already been given. However, unlike the present paper, they expanded around the treelevel definition of the VEV, as discussed in the Introduction above. This means that even at 1loop order, the results take different forms. The expression for the 1loop pole squared mass contribution given in eq. (2.22) above appears to differ from the result of eq. (B.4) of ref. Jegerlehner:2001fb and eq. (B.3) of ref. Jegerlehner:2002em by an amount
(2.34) 
in the notation of the present paper. There is of course no contradiction; this merely reflects the difference between the treelevel contributions, which are in this paper and in refs. Jegerlehner:2001fb ; Jegerlehner:2002em . Note in particular the presence of in eq. (2.34); at loop order , the use of the treelevel VEV results in terms proportional to . In contrast, there are no singularities in the present paper. A detailed comparison would be much more difficult at 2loop order, as refs. Jegerlehner:2001fb ; Jegerlehner:2002em also relied on doing highorder expansions in and and .
Iii Numerical results
Consider a benchmark set of Standard Model parameters defined at the input renormalization scale GeV:
(3.1)  
(3.2)  
(3.3)  
(3.4)  
(3.5)  
(3.6) 
The gauge couplings and are taken to agree with ref. Degrassi:2014sxa , while and are from eqs. (57) and (60) of version 4 of ref. Buttazzo:2013uya . The VEV and the Higgs selfcoupling were then chosen so that agrees with the central value of eq. (1.3), when computed at , and agrees with the current experimental central value Aad:2015zhl of GeV, when computed at GeV using the program SMH SMHwebpages as described in ref. Martin:2014cxa . With this set of input parameters, one also obtains from minimization of the Higgs potential using SMH at . In this way, the experimental measurements of and can be used to obtain the Higgs potential parameters. The choice of GeV for computing was explained in ref. Martin:2014cxa ; at this scale the effects of topquark loops in the neglected electroweak 3loop parts should be not too large. The lower choice of for computing is somewhat arbitrary. One also obtains a boson pole mass of GeV, when computed at , using the calculation described in Martin:2015lxa . This translates into a BreitWigner mass of GeV, using the analog of eq. (1.2) above. (Somewhat coincidentally, this agrees with the value found in ref. Degrassi:2014sxa to within 1 MeV, although that calculation uses a different scheme.)
The dependences of the computed pole mass parameters and on the choice of are shown in figures 3.1 and 3.2, in various approximations.
These graphs are made by running the input parameters , , , , , and , using their 3loop beta functions Chetyrkin:2013wya ; Bednyakov:2013eba , from the input scale to the scale on the horizontal axis, where is computed. In an idealized case that is computed to sufficiently high order in perturbation theory, and would be independent of . Therefore the independence is a check on the calculation. I find that the calculated 2loop value of varies by only about MeV from its median value, over the range 70 GeV 200 GeV. Below GeV, the scale dependence is much stronger. The scale dependence is smallest for near 100 GeV, where the computed has its minimum, but this does not necessarily mean that this is the best renormalization scale; only a higherorder calculation can reduce the theoretical uncertainty.
With regard to the width , the scale dependence of the full 2loop result is again about MeV from the median value over the same range 70 GeV 200 GeV. Note that here, including only the QCD part of the 2loop contribution does not actually reduce the scale dependence much compared to the 1loop result. This is because most of the dependence in the width arises from the runnings of the VEV and the electroweak couplings of the boson to the fermions into which it decays, and these are independent of QCD at the leading (1loop) order. The result for is consistent with, and slightly lower than the central value of, the experimental range RPP GeV. Of course, there are much better ways to calculate , because the imaginary part of the 2loop complex pole mass really corresponds to only a 1loop calculation of the width. (Moreover, the inclusion of bottomquark mass effects, neglected above for simplicity, has a larger effect on than on , and will decrease the former by an amount of order 2 MeV due to kinematics. There is a significant uncertainty in estimating this reduction in the imaginary part of the complex pole mass, because of the large difference between the pole and running bottom quark masses.)
It is important to keep in mind that the renormalization scale dependence only provides a lower bound on the theoretical error. Another way of investigating the robustness of the calculation is to take the running topquark squared mass in the 1loop part eq. (2.22) and perform an expansion around an arbitrary value that can be considered to differ from by an amount that is parametrically of 1loop order. An obvious choice is to take to be the (real part of the) topquark pole squared mass. It makes sense to do this in particular for the 1loop contribution, because the top quark mass appears only in propagators at this order, not as a vertex Yukawa coupling. Expanding, one finds:
(3.7)  
(3.8) 
I have checked that if these expansions were continued to include order , then the results for the pole squared mass would be nearly indistinguishable from the original result obtained directly from the unexpanded and . However, by instead keeping the expansion only at first order in as shown, one obtains an alternative consistent 2loop order result for the pole squared mass, since is to be treated as formally of 1loop order. This alternative consistent 2loop order result is numerically different, with the difference giving an indication of the magnitude of the error made in terminating perturbation theory at 2loop order. The result of using eqs. (3.7) and (3.8) compared to the original unexpanded and is shown in Figure 3.3.
We see that the alternate consistent 2loop result, shown as the dashed line, has a significantly worse scale dependence, especially at larger . This suggests that the scale dependence of found in the original calculation (the solid line) is actually accidentally small, and probably underestimates the theoretical error. A very similar behavior was found for the boson mass in ref. Martin:2015lxa .
Iv Outlook
In this paper I have provided a full 2loop calculation of the boson complex pole square mass in the pure scheme, to go along with similar results for the boson Martin:2015lxa and the Higgs boson Martin:2014cxa using the same renormalization scheme and the same definition of the VEV. These calculations are an alternative to the onshell scheme results that have been widely used for precision studies in the Standard Model, in which instead plays the role of an input parameter.
The ultimate goal should be to obtain results in which the theoretical error is very small compared to present and projected experimental errors. The previous section shows that this is certainly not obtained using just the full 2loop calculation, as the scale dependence is comparable to the experimental errors, and the theoretical error is probably somewhat larger. There is no compelling evidence or argument that the subset of 3loop contributions that are QCD and topYukawa enhanced will be enough to ensure the dominance of experimental errors over theoretical errors. At 2loop order, one can see from the benchmark example of Figure 3.1 that the QCD contribution has a much larger scale dependence, but not a much larger magnitude, than the nonQCD contributions, except for smaller choices of the renormalization scale where the topenhanced QCD corrections are big. The same thing was noted in the comparable results for the boson in Martin:2015lxa . It is therefore reasonable to conclude that complete 3loop calculations will be necessary, providing a worthy challenge for future work.
Acknowledgments: This work was supported in part by the National Science Foundation grant number PHY1417028.
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The source code for the program SMH is available at:
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