Dimensional Reduction applied to QCD at higher orders

Robert Harlander^{1}, Philipp Kant^{2}, Luminita Mihaila^{2}, Matthias Steinhauser^{2} ^{∗}

SFB/CPP-07-32, TTP/07-13, WUB/07-05 – June 2007

1- Fachbereich C - Bergische Universität Wuppertal 42097 Wuppertal - Germany 2- Theoretische Teilchenphysik - Universität Karlsruhe 76128 Karlsruhe - Germany

Recent developments in higher order calculations within the framework of Dimensional Reduction, the preferred regularization scheme for supersymmetric theories, are reported on. Special emphasis is put on the treatment of evanescent couplings, the equivalence to Dimensional Regularization, and the evaluation of α_{s}(M_{GUT}) from α_{s}(M_{Z}).

1 Dimensional Reduction

Dimensional Regularization (DREG) [?] has proven extremely successful for the evaluation of higher order corrections in quantum field theory, mostly because it preserves gauge invariance and thus does not interfere with the renormalizability of the Standard Model or QCD. Many techniques for evaluating Feynman diagrams have been developed within the framework of DREG, so perturbation theory heavily relies upon the validity of this regularization method.

Applied to SUSY theories, however, one faces the problem of explicit SUSY breaking by the need to assign different numbers of degrees of freedom to spin-1 and spin-1/2 fields. A manifestation of this SUSY breaking is that SUSY relations of couplings no longer hold at higher orders. For example, while SUSY requires equality for the quark-quark-gluon and the squark-quark-gluino couplings g and ĝ at all energy scales, one finds that their renormalization constants differ. In fact, it is Z_{g} = (1 + δĝ) Zĝ, and thus

ĝ = (1 + δĝ)g ,  (1)

where δĝ = α_{s}/(3π) [?]. In effect, the number of renormalization constants in SUSY becomes rather large when calculations are done in DREG.

As a way out, it was suggested to use Dimensional Reduction (DRED) as a regularization procedure for SUSY theories [?]. Formally, this means that space-time is compactified to D = 4 − 2ϵ dimensions (ϵ > 0), while the vector fields are kept four-dimensional. As an example, consider the electron-photon vertex, which in DRED becomes

ψ̄γ_{μ}ψA^{μ} = ψ̄γ_{μ}ψÂ^{μ} + ψ̄γ_{μ}ψÃ^{μ} = ψ̄γ̂_{μ}ψÂ^{μ} + ψ̄γ̃_{μ}ψÃ^{μ} ,  (2)

where Â^{μ} and Ã^{μ} denote the D and the 2ϵ-dimensional component of the vector field A^{μ}. Ã^{μ} is also called the ϵ-scalar. Traces over the D- and 2ϵ-dimensional γ-matrices can be evaluated using

{γ^{μ}, γ^{ν}} = 2g_{μν}  and  {γ̂^{μ}, γ̃^{ν}} = 0 .  (3)

Thus, perturbative calculations in DRED require to introduce additional fields (ϵ-scalars) and an extra set of γ-matrices. Once the algebraic part of the evaluation of a Feynman amplitude is done, the tools developped for DREG can be applied without further modification.

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∗Work supported by the DFG through SFB/TR 9 and HA 2990/3-1.

2 Evanescent couplings

In a SUSY theory, the relation A_{μ} = Â_{μ} + Ã_{μ} is essential: it is A_{μ} that is part of a supermultiplet, while Â_{μ} and Ã_{μ} are introduced for purely technical reasons. Therefore, this relation must hold also at higher orders of perturbation theory.

This is not necessarily the case in a non-SUSY theory. Since Ã_{μ} transforms like a scalar under gauge transformations (thus the name “ϵ-scalar”), there is no symmetry to ensure that the Ã_{μ}-couplings renormalize in the same way as the corresponding Â_{μ}-couplings. Thus, when applying DRED to QCD, we have to introduce two different couplings for the quarkgluon vertex, for example:

g_{s}A_{μ}ψ̄γ^{μ}ψ → ĝ_{s}Â_{μ}ψ̄γ̂^{μ}ψ + g̃_{s}Ã_{μ}ψ̄γ̃^{μ}ψ .  (4)

In order to be consistent with our journal papers [?, ?, ?]^{a} let us define

α_{s} = \frac{ĝ_{s}^{2}}{4π} ,  α_{e} = \frac{g̃_{s}^{2}}{4π} ,  (5)

where α_{e} is called “evanescent coupling”. Only at tree-level can one require that α_{s} = α_{e}. Higher orders lead to an energy dependece of the (minimally subtracted) couplings, governed by the RGEs^{b}

μ^{2} \frac{d}{dμ^{2}} α_{s} = β\overline{DR} ; s(α_{s}, α_{e}) ,  μ^{2} \frac{d}{dμ^{2}} α_{e} = β_{e}(α_{s}, α_{e}) .  (6)

The β-functions have been evaluated in Ref. [?] through three loops, and β\overline{DR} ; s is even known to four-loop order [?]. Indeed it turns out that β\overline{DR} ; s ≠ β_{e} in standard QCD already at one-loop level. The condition α_{s} = α_{e} can therefore be implemented only at one particular value of μ^{2}.

If \overline{DR} (i.e., DRED with minimal subtraction) is to be a viable renormalization scheme, then one should be able to transform physical results from one scheme into the other by finite shifts of the renormalized parameters. This property has been confirmed several times [?, ?, ?]. The proper conversion relation for the strong coupling between the \overline{MS} and the \overline{DR} scheme in n_{f}-flavor standard QCD is given at two-loop level by [?]

ᾱ_{s} = α_{s} [1 − \frac{α_{s}}{4π} − \frac{5}{4} (\frac{α_{s}}{π} )^{2} + \frac{α_{s}α_{e}}{12π^{2}} n_{f} + …] ,  (7)

where ᾱ_{s} denotes the strong coupling in the \overline{MS} scheme. Three-loop corrections to this relation are known as well [?].

When evaluating physical observables in DREG, the result depends only on ᾱ_{s}, while it depends on both α_{s} and α_{e} in DRED. This ambiguity should be viewed as a freedom of the renormalization scheme: any choice of α_{e} determines the value of α_{s} by comparison to the experimental value of the physical observable at one particular scale μ_{0}. At any other scale μ, α_{s} and α_{e} are determined by the RGEs Eq. (6).

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aThere is a misprint in Eq. (18) of Ref. [?]: the term −25 n_{f}^{2} /72 should read −25ζ_{3}/72.

bIn fact, there are several evanescent couplings in QCD; however, for the sake of the argument, it is sufficient to consider only α_{e} here.