Supersymmetry. Theory, Experiment, and Cosmology

400 A review of the Standard Model and of various notions of quantum field theory

One may distinguish three types of radiative corrections in the Standard Model:

(i)α and Gµ are measured at low energy and must be renormalized up to the scale

at which present measurements are performed (typically MZ ). For example, summing the large (ln MZ /me)n contributions into a running coupling α(MZ ), one obtains from the boundary value (A.172) at a scale me

where the error quoted is mainly due to the uncertainty on the low energy hadronic contribution to vacuum polarization.

On the other hand, Gµ does not receive large ln (MZ /mµ) corrections because of

anonrenormalization theorem12.

(ii)Large corrections due to the top or the Higgs may appear through vacuum polarization diagrams. These are traditionally called oblique corrections. There is

awell-defined procedure to take them into account.

One may introduce in general the two point-function:

µ q q ν

Since this is usually contracted with external fermionic currents, the terms O (qµqν ) give contributions of the order of the light fermion masses, which we neglect (see however below). If we denote by M the mass of the heavy field (Z, top or Higgs), then dimensional analysis tells us that A M 2, F M 0 and G M −2 and thus the term in q4 is negligible for q2 M 2.

We thus have to consider

where Aγγ = AγZ = 0 because the electromagnetic current is conserved: qµΠγγµν = 0 = qµΠγZµν for q2 = 0.

12Through Fierz reordering, one may write the e ective current–current interaction responsible for muon decay as

Gµ

Le = √

2

The lepton-changing (e → µ) vector and axial currents that appear satisfy a nonrenormalization theorem that allows only finite corrections and hence forbid any dependence in the e ective cut-o MZ .

Electroweak precision tests 401

Fig. A.8

Now, if we consider the diagrams of Fig. A.8, only the tree-level and γγ propagator contribute to the pole in q2. Thus the pole part of the amplitude reads

where we have introduced, as in (A.186), the bare e0 and renormalized e couplings. Thus, to first order, δe2/e2 = −Fγγ .

Thus, three out of six of the independent quantities that we have introduced in (A.193) are used to renormalize the variables α, Gµ and MZ . We are left with three variables which fully describe the oblique corrections [6, 307].

One prefers to work in the original SU (2) × U (1) basis (A3µ, Bµ) for the gauge fields. One then introduces13:

13Alternatively, in the language of [307, 308], one introduces the three variables S, T and U which,

Dilatations and renormalization group 403

where M ≡ h + iτ · ξ. This Lagrangian has an invariance SU (2)L × SU (2)R: M → ULM UR† , where UL, UR are 2 × 2 unitary matrices (to understand the meaning of the L, R subscripts, see Equation (7.9) of Chapter 7). When we restore the gauge fields, there remains a SU (2)R invariance in the limit θW → 0 (g → 0). When mH → ∞, the model becomes nonlinear with the constraint Tr M †M = v2. The SU (2)R symmetry still constrains the structure of the counterterms and forbids terms of the form GF m2H (mH is the physical cut-o ).

This result is known as the screening theorem [351]. The sensitivity of low energy physics in the Higgs mass is only logarithmic. But tests at the LEP collider have achieved such a precision that they now allow us to put limits on the Higgs mass within the Standard Model.

(iii) In the case of processes involving the third generation of quarks, one has a possible

→ ¯ large dependence in mt through vertex corrections (e.g. Z bb).

A.5 Dilatations and renormalization group

Scale invariance and its violations as described by the renormalization group approach play an important rˆole in the study of supersymmetry. We present the basic notions in this section. We also review the notion of e ective potential which is used in the main text to discuss quantum corrections.

A.5.1 Dilatations and conformal transformations

A dilatation or scaling transformation is a spacetime transformation of the form

The number d is characteristic of the field Φ and is called its scaling dimension. Infinitesimally,

At the classical level, the scaling dimension coincides with the canonical dimension: d = 1 for the scalar fields, 32 for spin 12 or 32 fermions.

We may consider as an example the following action involving a scalar field φ(x) and a Dirac spinor field Ψ(x):

Dilatations and renormalization group 405

dilatation current (A.222) is expressed by the condition of tracelessness for the new energy–momentum tensor:

It then follows that the following four (λ = 0, 1, 2, 3) currents are conserved:

These are actually the four currents associated with the four generators of conformal transformations Kλ ≡ d4xKλ0. The corresponding transformations are obtained by combining the four translations xµ → xµ − aµ with the inversion xµ → −xµ/x2:

We note that x µ/x 2 = xµ/x2 + cµ. By construction, an invariant line element transforms as

The possibility of defining a tensor Θµν is associated with the presence of conformal invariance in the limit ∆ → 0. One may note here the close link between conformal and dilatation invariance. Thus, any Lorentz invariant theory which is also invariant under conformal transformations has dilatation invariance. This can be seen from the general relation

which complements the Poincar´e algebra (given in (2.10) of Chapter 2) to form the algebra of the conformal group. According to the last relation, invariance under Pµ, Mµν , and Kµ implies dilatation invariance.

We end this section with a few words on the conformal group in D dimensions. As we just saw in (A.227) for the case D = 4, the conformal group is the subgroup of coordinate transformations or reparametrizations such that the metric transforms as14

This group obviously includes the Poincar´e group since it leaves the metric invariant.

14This transformation preserves the angle between two vectors aµ and bµ: gµν aµbν/ gµν aµaν gρσ bρbσ .

408 A review of the Standard Model and of various notions of quantum field theory

Γφ is obtained by a Legendre transformation:

as can be verified explicitly by functionally di erentiating this relation with respect to φ(x).

The functional Γ φ is called the e ective action. Two of its properties will be of

values of the fields in the fundamental state of the interacting theory (the vacuum |Ω ). This turns out to be useful when studying the spontaneous breaking of a symmetry. If this aspect alone is of interest to us, one may restrict one’s attention to the nonderivative terms in the e ective action which define the e ec-

(ii)The e ective action is also the generating functional of proper Green’s functions (i.e. one-particle irreducible and truncated Green’s functions). In momen-

yields proper Green’s functions. This allows us to compute the e ective action from truncated Green’s functions. In particular, the e ective potential corresponds to vanishing momentum on external legs:

The e ective action is usually computed in powers of h¯, which corresponds to an expansion in the number L of loops. Let us illustrate, in the example of a theory in λφ4, how one determines the e ective potential. We consider the theory of a real massless scalar field with action:

where we have included counterterms in the second line.