SFB/CPP-07-31 TTP/07-12 WUB/07-04

Running of α_{s} and m_{b} in the MSSM

R.V. Harlander^{(a)}, L. Mihaila^{(b)}, M. Steinhauser^{(b)} (a) Fachbereich C, Theoretische Physik, Universität Wuppertal, 42097 Wuppertal, Germany (b) Institut für Theoretische Teilchenphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany

Abstract

A consistent evolution of the strong coupling constant α_{s} from M_{Z} to the GUT scale is presented, involving three-loop running and two-loop decoupling. The twoloop transition from the \overline{MS}- to the \overline{DR}-scheme is properly taken into account. In the second part of the paper, the bottom quark mass in the \overline{DR} scheme at the electroweak/SUSY scale is evaluated with four-loop accuracy. We find that the three-loop effects are comparable to the experimental uncertainty both for α_{s} and m_{b}.

PACS numbers: 11.30.Pb, 12.38.-t, 12.38.Bx, 12.10.Kt

1 Introduction

The Minimal Supersymmetric Standard Model (MSSM) is one of the most studied extensions of the Standard Model (SM). The observation that the gauge couplings of the strong, electromagnetic and weak interaction tend to unify in the MSSM at a high energy scale μ_{GUT} ≃ 10^{16} GeV [?,?,?,?,?] consistent with Grand Unification Theories (GUT) is probably the most compelling hint in favour of supersymmetry (SUSY). Constraints from the Yukawa sectors of SUSY-GUT models yield appealing predictions for SM parameters, including the top quark mass and the ratio of the bottom quark to the tau lepton masses [?,?,?,?].

Nevertheless, SUSY can only be an approximate symmetry in nature. Several scenarios for the mechanism of SUSY breaking, based on very different physical models, have been proposed. A possibility to constrain the type and scale of SUSY breaking is to study, with very high precision, the relations between the MSSM parameters evaluated at the electroweak and the GUT scales. The extrapolation over many orders of magnitude requires high-precision experimental data at the low energy scale. A first set of precision measurements is expected from the CERN Large Hadron Collider (LHC) with an accuracy at the percent level. A comprehensive high-precision analysis can be performed at the International Linear Collider (ILC), for which the estimated experimental accuracy is at the per mill level. It is obvious that the same precision must be reached also on the theory side in order to match with the data [?]. Running analyses based on full twoloop renormalization group equations (RGEs) [?, ?, ?, ?] for all parameters and one-loop threshold corrections [?] are currently implemented in the public programs ISAJET [?], SOFTSUSY [?], SPHENO [?], SuSpect [?]. The agreement between the different codes is in general within one percent [?]. A first three-loop running analysis, based, however, only on one-loop threshold effects, was carried out in Ref. [?, ?]. While the three-loop running has little effect on the weakly interacting particle sector of the MSSM, this effect can be comparable to that of two loop running for strongly interacting particles, as for example the squark mass spectrum [?].

In this paper, we elaborate an earlier investigation [?] of the evolution of the strong coupling α_{s} in MSSM, based on three-loop RGEs [?] and two-loop threshold corrections. On the one hand, the three-loop corrections reduce significantly the dependence on the scale at which heavy particles are integrated out and on the other hand, they are essential for phenomenological studies, because they are as large as, or greater than, the effects induced by the current experimental accuracy of α_{s}(M_{Z}). Our aim is to compute α_{s} at a high-energy scale μ ≃ O(μ_{GUT}) with three-loop accuracy, starting from the SM parameter α_{s}(M_{Z}) as input. Additionally, we compare the predictions obtained within the above mentioned approach with those based on the leading-logarithmic (LL) approximation suggested in Ref. [?].

In the context of SO(10) GUT [?, ?] the third-family Yukawa couplings may unify for a large ratio of the Higgs vacuum expectation values tan β ≃ 50. Moreover, in all SUSY models with large tan β, the supersymmetric particle spectrum and the Higgs boson masses are sensitive to the bottom Yukawa coupling [?, ?]. In turn, the relation between the running bottom mass and the bottom Yukawa coupling is affected in such theories by large supersymmetric radiative corrections [?]. For the calculation of higher order corrections in the framework of SUSY theories, the \overline{DR} scheme, based on regularization by dimensional reduction and modified minimal subtraction, is most convenient. On the other hand, the values of the SM parameters are presently extracted from the experimental data for the \overline{MS} renormalization scheme. It is obvious that translation formulae from one scheme to the other to the appropriate order in the perturbative expansion have to be employed for phenomenological studies.

Owing to these considerations, a precise determination of the running bottom mass in the \overline{DR} scheme is of great importance for SUSY theories sensitive to the bottom quark Yukawa coupling. Ref. [?] proposed an approach to the determination of m\overline{DR} ; b(M_{Z}), which avoids the infrared sensitivity problem related with the use of the bottom-quark pole mass. The procedure consists of employing the RGEs of QCD and the relation between \overline{MS} and \overline{DR} masses in order to calculate m\overline{DR} ; b, using as input parameter the accurately known μ_{b} = m\overline{MS} ; b(μ_{b}). A second purpose of this paper is to extend the study of Ref. [?] to three- and even to four-loop order [?, ?], in order to take full advantage of the recent determination of m\overline{MS} ; b(μ_{b}) with four-loop accuracy [?]. Furthermore, we study the phenomenological significance of evanescent couplings related with the application of the \overline{DR} scheme to QCD.

The remainder of the paper is organized as follows: in Section 2 we discuss the evaluation of the strong coupling at the GUT scale from the knowledge of α_{s}(M_{Z}) and propose a consistent prescription based on three-loop running and two-loop decoupling. Section ?? deals with the running of the bottom quark mass from μ = μ_{b} to a decoupling scale of supersymmetric particles.

2 Running and decoupling of α_{s}

The value of the strong coupling constant, measured at the mass of the Z boson M_{Z}, is a central quantity in high energy physics. Usually, what is being quoted in the literature as α_{s}(M_{Z}) is its value in a theory with five quark flavours, renormalized in the \overline{MS} scheme. In order to be precise, we will refer to this quantity as α_{s}^{\overline{MS},(5)}(M_{Z}).

On the contrary, what is meant by the value of α_{s}(μ_{GUT}) is the strong coupling constant in a supersymmetric theory, renormalized in the \overline{DR} scheme. We will denote this quantity by α_{s}^{\overline{DR},(full)}(μ_{GUT}) in what follows. It is the purpose of this section to study the relation between α_{s}^{\overline{MS},(5)}(M_{Z}) and α_{s}^{\overline{DR},(full)}(μ_{GUT}) at three-loop level. This requires the consistent combination of (a) the renormalization group evolution of α_{s}; (b) the transition from the \overline{MS} to the \overline{DR} scheme; (c) the transition from five-flavour QCD to the full SUSY theory. All these issues will be discussed in detail in what follows.

2.1 Renormalization group evolution

The energy dependence of the strong coupling constant is governed by the RGE

μ^{2} \frac{d}{dμ^{2}} α^{s}(μ^{2}) = β(α^{s}) ,  β(α^{s}) = −α2 s ∑_{n≥0} β^{n}αn s .  (1)

The β function depends on the underlying theory and on the renormalization scheme. In QCD with n_{f} quark flavours, it is known in the \overline{MS} scheme through four loops [?]; the first three coefficients read

\begin{aligned}β_{0} & = \frac{1}{4} (11 − \frac{2}{3}n_{f}) ,  β_{1} = \frac{1}{16} (102 − \frac{38}{3} n_{f}) , \\ β_{2} & = \frac{1}{64} (\frac{2857}{2} − \frac{5033}{18} n_{f} + \frac{325}{54}n_{f}^{2}) . \end{aligned}  (2)

The QCD β function in the \overline{DR} scheme involves evanescent couplings (see Section 2.3 below) at three loops and higher. Explicit results through four loops are given in Refs. [?,?].

In SUSY-QCD, the β function has been evaluated in the \overline{DR} scheme through three loops from the NSVZ formula [?,?] by a proper shift in the strong coupling constant [?].

For six (s)quark flavours one finds

β_{0} = \frac{3}{4} ,  β_{1} = −\frac{7}{8} ,  β_{2} = −\frac{347}{192} .  (3)

2.2 Decoupling of the SUSY particles

For mass independent renormalization schemes like \overline{MS} or \overline{DR}, the decoupling of heavy particles has to be performed explicitely. In practice, this means that intermediate effective theories are introduced by integrating out the heavy degrees of freedom. The parameters of the various effective theories are related by so-called decoupling (or matching) constants.

When going from a high to a low energy scale, one may separately integrate out every particle at its individual threshold. This multi-scale approach (sometimes denoted as “step approximation”) is suited for SUSY models with a severely split mass spectrum [?]. As pointed out long time ago, the intermediate effective theories with “smaller” symmetry raise the problem of introducing new couplings, each governed by its own RGE. This is the case in the Yukawa sector if the SU(2) symmetry is broken, for example. It also happens for the “evanescent” couplings, related with the application of DRED to non-supersymmetric theories. A comprehensive multi-scale decoupling procedure in the context of the MSSM is currently not available, although it could be of phenomenological relevance [?].

On the other hand, in SUSY models with roughly degenerate mass spectrum at a scale M̃, one can consider the MSSM as the full theory that is valid from the GUT scale μ_{GUT} down to M̃, which we assume to be around 1 TeV. Integrating out all SUSY particles at this common scale, one directly obtains the SM as the effective theory, valid at low energies. As already mentioned above, the strong coupling constants in SUSY and in QCD are then related by a decoupling constant ζ_{s}. The transition between the two theories can be done at an arbitrary decoupling scale μ:^{1}

α_{s}^{\overline{DR},(n_{f})}(μ) = ζ_{s}^{(n_{f})} α_{s}^{\overline{DR},(full)}(μ) .  (4)

ζ_{s} depends logarithmically on the scale μ, which is why one generally chooses μ ∼ M̃. In Eq. (4), n_{f} = 6 means that only the SUSY particles are integrated out, while for n_{f} = 5 at the same time the top quark is integrated out. This procedure, also known as “common scale approach” [?], has the advantage that it avoids the occurrence of intermediate non-supersymmetric effective theories. Most of the present codes computing the SUSY spectrum [?, ?, ?] follow this approach by applying the one-loop approximation of Eq. (4) and setting n_{f} = 5 and μ = M_{Z}.

In this paper, we apply the corresponding two-loop approximation, as it is required by a consistent three-loop evolution of the coupling constant [?]. Furthermore, in contrast to most analyses performed up to now, we allow for a general decoupling scale μ = μ_{dec} where the heavy particles are decoupled.

1In principle the decoupling constants should also carry a label \overline{MS} or \overline{DR}. However, in this paper decoupling is always performed in the \overline{DR} scheme, and therefore we omit the label.

The decoupling coefficients of Eq. (4) have been evaluated through two loops in Eqs. (26) and (29) of Ref. [?]:

ζ_{s}^{(n_{f})} = 1 + \frac{α_{s}}{π} ζ_{s1}^{(n_{f})} + (\frac{α_{s}}{π} )^{2} ζ_{s2}^{(n_{f})} + O(α_{s}^{3}) ,  (5)

where α_{s} ≡ α_{s}^{\overline{DR},(full)}(μ), and

\begin{aligned}ζ_{s1}^{(6)} & = −LM̃ ,  ζ_{s1}^{(5)} = −\frac{1}{6}L_{t} − LM̃ , \\ ζ_{s2}^{(6)} & = −\frac{65}{32} − \frac{5}{2}LM̃ + L2 ; M̃ , \\ ζ_{s2}^{(5)} & = −\frac{215}{96} − \frac{19}{24}L_{t} − \frac{5}{2}LM̃ + [\frac{1}{6}L_{t} + LM̃]^{2} \\ & + (\frac{m_{t}}{M̃} )^{2} (\frac{5}{48} + \frac{3}{8}L_{tM̃}) − \frac{7π}{36} (\frac{m_{t}}{M̃} )^{3} \\ & + (\frac{m_{t}}{M̃} )^{4} (\frac{881}{7200} − \frac{1}{80}L_{tM̃}) + \frac{7π}{288} (\frac{m_{t}}{M̃} )^{5} + …. \end{aligned}  (6)

M̃ is the common mass of the SUSY particles, m_{t} is the top quark mass, and

LM̃ = ln \frac{μ^{2}}{M̃^{2}} ,  L_{t} = ln \frac{μ^{2}}{m_{t}^{2}} ,  L_{tM̃} = ln \frac{m_{t}^{2}}{M̃^{2}} .  (7)

Although the formula for ζ_{s2}^{(5)} in Eq. (6) has been obtained in the limit m_{t} ≪ M̃, it has been shown that it holds at the per cent level for all values of m_{t} ≤ M̃ [?]. The exact limit m_{t} = M̃ is also given in Ref. [?].

2.3 \overline{MS}–\overline{DR} conversion and evanescent couplings

In order to use Eq. (4), one needs to transform the input value for α_{s} from the \overline{MS} to the \overline{DR} scheme. The corresponding relation has been evaluated through three loops in Ref. [?]. Here, we only need the two-loop expression [?]:

α\overline{MS} ; s = α\overline{DR} ; s [1 − \frac{α\overline{DR} ; s}{4π} − \frac{5}{4} (\frac{α\overline{DR} ; s}{π} )^{2} + \frac{α\overline{DR} ; sα_{e}}{12π^{2}} n_{f} + …] ,  (8)

where α\overline{DR} ; s ≡ α_{s}^{\overline{DR},(n_{f})}(μ) and α\overline{MS} ; s ≡ α_{s}^{\overline{MS},(n_{f})}(μ).

α_{e} ≡ α_{e}^{(n_{f})}(μ) is one of the so-called evanescent coupling constants that occur when DRED is applied to non-supersymmetric theories (QCD in this case). In particular, it describes the coupling of the 2ε-dimensional components (so-called ε-scalars) of the gluon to a quark. Other evanescent couplings that occur in QCD are η_{r} (r = 1, 2, 3), related to the vertex of four ε-scalars (see, e.g., Ref. [?] for details). In the following, we will assume that QCD is obtained by integrating out the heavy degrees of freedom (squarks and gluinos) from SUSY-QCD. In this case, the evanescent couplings can be related to the gauge coupling α_{s} as follows:^{2}

α_{e}^{(full)}(μ) = α_{s}^{\overline{DR},(full)}(μ) = η_{1}^{(full)}(μ) ,
η_{2}^{(full)}(μ) = η_{3}^{(full)}(μ) = 0 .  (9)

The evanescent couplings in n_{f}-flavour QCD, i.e. α_{e}^{(n_{f})} and η_{r}^{(n_{f})} are then obtained by decoupling relations analogous to Eq. (4).

2.4 Decoupling for α_{e}

For the one-loop decoupling relation that connects the values of α_{e} in five-flavour QCD and in SUSY-QCD, we find by a calculation similar to Ref. [?]:

α_{e}^{(5)}(μ) = ζ_{e}α_{e}^{(full)}(μ) ,
ζ_{e} = 1 + \frac{α_{s}^{\overline{DR},(full)}(μ)}{π}[T_{F} (−\frac{1}{2}L_{t}) + C_{A} (−\frac{1}{4}Lg̃ + \frac{(m2 ; g̃(1 + Lg̃) − m2 ; q̃(1 + Lq̃))}{2(m2 ; g̃ − m2 ; q̃)})

+C_{F} (\frac{m2 ; q̃ − 3m2 ; g̃}{4(m2 ; g̃ − m2 ; q̃)} + \frac{m2 ; q̃(2m2 ; g̃ − m2 ; q̃)Lq̃ − m4 ; g̃Lg̃}{2(m2 ; g̃ − m2 ; q̃)^{2}}) ] + O(α_{s}^{2}) ,  (10)

where we have used the short-hand notation

L_{x} = ln \frac{μ^{2}}{m_{x}^{2}} ,  x ∈ \{t, g̃, q̃\} .  (11)

Furthermore, the mixing angle between the squark mass eigenstates m_{q̃1} and m_{q̃2} has been set to zero. Note that Eq. (10) depends on a specific (s)quark flavour q. This means that actually ζ_{e} and therefore also α_{e} should carry an additional label q:

α_{e}^{q,(5)}(μ) = ζ_{e}^{q} α_{e}^{(full)}(μ) .

For example, α_{e}^{u,(5)} is the coupling constant for the ūuε vertex, where the decoupling constant ζ_{e}^{u} depends on mũ. Therefore, if all squark masses are different, one needs a separate evanescent coupling for each quark flavour. We remark that gauge invariance prevents the occurrence of such a flavour dependence for α_{s}^{\overline{DR},(5)}. If all squarks have the same mass mq̃, then all evanescent couplings are the same, α_{e}^{u,(5)} = α_{e}^{d,(5)} = ⋯ = α_{e}^{b,(5)}, and one can drop the flavour label. In the limit mg̃ = mq̃ = M̃, Eq. (10) reads

ζ_{e} = 1 + \frac{α_{s}^{\overline{DR},(full)}(μ)}{π}[T_{F} (−\frac{1}{2}L_{t}) + C_{A} (\frac{1}{4}LM̃) + C_{F} (−\frac{1}{2}LM̃) ] .  (12)

2Assuming that DRED preserves supersymmetry.

α_{e} occurs typically at higher orders, and in fact, as we will now argue, one can easily eliminate it perturbatively from the three-loop running of α_{s}.

Because of Eq. (9), one can write

α_{e}^{(n_{f})}(μ) = ξ^{(n_{f})} α_{s}^{\overline{DR},(n_{f})}(μ) ,  (13)

with

ξ^{(n_{f})} = \frac{ζ_{e}^{(n_{f})}}{ζ_{s}^{(n_{f})}} = 1 + \frac{α_{s}^{\overline{DR},(n_{f})}(μ)}{π} ξ_{1}^{(n_{f})} + O(α_{s}^{2}) ,  (14)

where the dependence on α_{s}^{\overline{DR},(full)} has been eliminated by using Eq. (4). In case all heavy particles have a common mass M̃, the quantity ξ_{1}^{(n_{f})} only depends on ln(μ^{2}/M̃^{2}) and thus for μ ≡ μ_{dec} ≈ M̃ no numerically enhanced perturbative coefficients appear. This is even true for a mass spectrum where all mass ratios are within the range of one or two orders of magnitude. This absence of artificially large logarithms allows us to write

α_{e}^{(n_{f})}(μ_{dec}) = α_{s}^{\overline{DR},(n_{f})}(μ_{dec}) + O(α_{s}^{2}) .  (15)

In this way, we can replace α_{e} by α\overline{DR} ; s in the two-loop term of Eq. (8).

2.5 Evaluation of α_{s}(μ_{GUT}) from α_{s}(M_{Z})

There are various ways to go from α_{s}^{\overline{MS},(n_{f})}(M_{Z}) to α_{s}^{\overline{DR},(full)}(μ_{GUT}). They most importantly differ in the degree to which evanescent couplings are needed. We propose the following method:

α_{s}^{\overline{MS},(n_{f})}(M_{Z}) →^{(i)} α_{s}^{\overline{MS},(n_{f})}(μ_{dec}) →^{(ii)} α_{s}^{\overline{DR},(n_{f})}(μ_{dec})

→^{(iii)} α_{s}^{\overline{DR},(full)}(μ_{dec}) →^{(iv)} α_{s}^{\overline{DR},(full)}(μ_{GUT}) .

(16)

The individual steps require

(i) β(α_{s}) in QCD through three loops [Eqs. (1), (2)]

(ii) the \overline{MS}–\overline{DR} relation through order α_{s}^{2} [Eq. (8)]; due to Eq. (15) one can set α_{e} = α\overline{MS} ; s at this order

(iii) decoupling through order α_{s}^{2} [Eqs. (4) with n_{f} = 5, Eq. (6)]; no evanescent couplings appear at this order (for a “common scale approach”). Note that if the decoupling of the supersymmetric particles were performed in several steps, α_{e} would appear in the two-loop decoupling coefficients

(iv) β(α_{s}) through three loops in SUSY [Eqs. (1), (3)] An advantage of this procedure as compared to a multi-scale approach is that the RGEs are only one-dimensional (there is only one coupling constant), and that for α_{e} one can apply Eq. (9) and Eq. (15).

Let us remark that in principle it is possible to decouple the top quark separately:

α_{s}^{\overline{MS},(5)}(M_{Z}) →^{(i′)} α_{s}^{\overline{MS},(5)}(μ_{t}) →^{(ii′)} α_{s}^{\overline{MS},(6)}(μ_{t}) →^{(iii′)} α_{s}^{\overline{MS},(6)}(μ_{dec}) (17)

and continue with steps (ii)–(iv) of Eq. (16), but now using n_{f} = 6 in all the formulas. μ_{t} is an additional decoupling scale to be chosen of the order of m_{t}. The only new ingredient needed is the decoupling constant for going from five to six quark flavours in the \overline{MS} scheme in step (ii′). It has been evaluated through three and four loops in Refs. [?] and [?,?], respectively. In any case, for a mass spectrum as given by the benchmark point SPS1a′ [?], for example, the separate decoupling of the top quark implies a numerically small effect. This can also be established by comparing “Scenario D” and “Scenario C” in Ref. [?].

For a direct application of Eqs. (4), (6), and (8) to steps (ii) and (iii) of Eq. (16), these equations need to be inverted. In fact, it may be convenient for practical phenomenological analyses to combine steps (ii) and (iii) of Eq. (16) into a single formula:

α_{s}^{\overline{DR},(full)} = α_{s}^{\overline{MS},(n_{f})}\{1 + \frac{α_{s}^{\overline{MS},(n_{f})}}{π} (\frac{1}{4} − ζ_{s1}^{(n_{f})})

+ (\frac{α_{s}^{\overline{MS},(n_{f})}}{π} )^{2} [\frac{11}{8} − \frac{n_{f}}{12} − \frac{1}{2} ζ_{s1}^{(n_{f})} + 2 (ζ_{s1}^{(n_{f})})^{2} − ζ_{s2}^{(n_{f})}] \} ,

(18)

where the scale μ_{dec} has been suppressed in the notation. The coefficients ζ_{s1}^{(n_{f})} and ζ_{s2}^{(n_{f})} are given in Eq. (6). The numerical deviation of Eq. (18) from the two-step procedure is well below 0.2%.

2.6 Numerical results

The result for α_{s}^{\overline{DR},(full)}(μ_{GUT} = 10^{16} GeV), obtained using M_{Z} = 91.1876 GeV and

m_{t} = 170.9 ± 1.9 GeV , α_{s}^{\overline{MS},(5)}(M_{Z}) = 0.1189 , M̃ = mq̃ = mg̃ = 1000 GeV (19)

as input parameters is shown if Fig. ??. The dotted, dashed and solid line are based on Eq. (16), where n-loop running is combined with (n − 1)-loop decoupling, as it is required for consistency (n = 1, 2, 3, respectively). We find a nice convergence when going from one to three loops, with a very weakly μ_{dec}–dependent result at three-loop order.

For comparison, we show the result obtained from the formula given in Eq. (21) of Ref [?]. It corresponds to the dash-dotted line in Fig. ??. In this case, the evolution is done in two steps:

α_{s}^{\overline{MS},(5)}(M_{Z}) ⟶^{(i)} α_{s}^{\overline{DR},(full)}(M_{Z}) ⟶^{(ii)} α_{s}^{\overline{DR},(full)}(μ_{GUT}) .  (20)