Supersymmetry. Theory, Experiment, and Cosmology

Naturalness and the problem of hierarchy 9

where α is a positive or negative number of order one. Taking Λ as a fundamental mass unit,

Plugging typical numbers, say m 100 GeV and Λ MP 1019 GeV, we see that m20/Λ2 must be adjusted to more than 30 orders of magnitude. This is to most people an intolerable fine tuning.

Let us be more precise in the case of the Standard Model. Then, the Higgs mass receives the following one-loop corrections:

that the one-loop quadratically divergent contribution vanishes if we have the following relation between the masses:

This relation, known as the Veltman condition [352], is obviously not ensured to hold at higher orders.

One may define the amount f of fine tuning discussed above by

Indeed, if δm2h = 100m2h, then one needs to fine tune the Higgs bare mass m20 to the per cent level in order to recover the right physical Higgs mass m2h. This amount of fine-tuning is represented on a plot (mh, Λ) [257] in Fig. 1.2 for values of Λ smaller than 100 TeV. The regions forbidden by the triviality and the vacuum stability bounds discussed in the preceding sections are also presented. One may note that, in the region corresponding to the Veltman condition (1.25), there is less need for fine tuning: this region does not extend, however, to very large values of Λ because of the higher order contributions.

In any case, it is clear that fine tunings larger than the per cent level are necessary as soon as the scale of new physics Λ is larger than 100 TeV.

Such a fine tuning goes against the prejudice that the observable properties of a theory (masses, charges,...) are stable under small variations of the fundamental parameters (the bare parameters). One talks of the naturalness of a theory to describe such behavior.

Let us see how this operates in quantum electrodynamics, which is the archetype of a natural theory. Since QED is characterized by a dimensionless coupling, one may wonder what is the fundamental scale. However, we are again in the situation of a “trivial” theory with an ever increasing coupling (at least perturbatively) and thus

Naturalness and the problem of hierarchy 11

For example, the electron mass is protected by the chiral symmetry: in the limit me → 0, the symmetry of the system is enhanced to include invariance under ψe → eiαγ5 ψe where ψe is the Dirac spinor describing the electron. The presence of this symmetry imposes that the corrections to the electron mass are themselves proportional to me. It follows that the coe cient of proportionality is dimensionless and thus behaves like log Λ.

This leads to the formulation of naturalness by ’t Hooft [349]:

A theory is natural if, for all its parameters p which are small with respect to the fundamental scale Λ, the limit p → 0 corresponds to an enhancement of the symmetry of the system.

1.2.3The case of the scalar field

Let us return to the case of a complex scalar field with self-coupling λ which was discussed in the previous section.

The fundamental high-energy scale of the theory is given again by its Landau pole ΛLandau given in (1.15). But not all parameters are naturally small at this scale:

•λ may be small because λ = 0 corresponds to an enhancement of the symmetry (conservation of the number of Φ particles).

•m2 is not naturally small because m2 = 0 does not correspond to any symmetry enhancement at the quantum level. To be more precise, in the limit λ → 0 and m2 → 0, the symmetry of the system is larger: Φ(x) → Φ(x)+C, with C constant. This could be the symmetry of a theory which would appear at a scale Λnat. At low energy this symmetry would be broken by e ects described by a parameter ε: λ = 0(ε), m2 = 0(ε) Λ2nat. Thus

m

Λnat √ .

λ

√

In other words, a Φ4 theory is natural for a fundamental scale of order m/ λ.

√

If one applies this to the Standard Model, one obtains m/ λ v. Thus, strictly speaking, the Standard Model is only natural up to a scale no larger than the TeV.

1.2.4The case of asymptotically free gauge theories. Technicolor

Contrary to scalar field theories, asymptotically free gauge theories provide good candidates for natural theories. In the limit of vanishing gauge coupling (free theory!), the conservation of the number of gauge bosons enhances the symmetry. One can easily verify that there is no severe problem of fine-tuning.

Denote by Λ the fundamental scale, where the gauge coupling is, in some sense, the bare coupling g0. Then keeping for simplicity only the first term in the beta function

At this scale, gauge interactions become strong and typically generate physical masses of this order. Taking the example of SU (3) (b = 11), one obtains ΛIR/Λ 10−19 for g02 = 0.16: no need to adjust g02 intolerably! This is precisely the property which is used in technicolor models to overcome the naturalness problem: scalars are bound states and m2scalar Λ2.

1.3Supersymmetry as a solution to the problem of naturalness

We have seen that:

(i)setting the mass of a scalar field to zero does not enhance the symmetry;

(ii)setting the mass of a fermion field to zero enhances the symmetry (chiral symmetry).

The idea is therefore to relate under a new symmetry a scalar field with a fermion field. This symmetry – supersymmetry – must be such that the masses of the scalar and of the fermion fields be equal. It is therefore related, in some sense to be defined, to the invariance under the Poincar´e group since it connects representations of di erent spin. In such a scheme, the relation ms/Λ 1 is natural because mf /Λ 1 is natural and because the scalar mass ms is related to the fermion mass mf .

Then, the contribution of fermions to the quadratic divergence cancels the contribution of bosons.

1.3.1An explicit example: the Wess–Zumino model

We will check this result explicitly, on a model known as the Wess–Zumino model. This model might seem at first rather contrived. We will learn how to construct it and to understand the beauty of it (see Chapter 3). For the time being, it will be an opportunity to familiarize ourselves with some typical supersymmetric interactions.

(ii)an interaction term which is expressed in terms of a single function W (φ) known as the superpotential

14 The problems of the Standard Model

which shows that the quadratic divergences (still present in the first two terms) cancel6.

One may show that the cancellation is even larger and that logarithmic divergences are only present in wave function renormalization. There are no mass counterterms and, more generally, the parameters of the superpotential are not renormalized (no finite or infinite quantum corrections). This is an example of the famous nonrenormalization theorems that have made the success of supersymmetry.

[The rest of this chapter is not necessary for reading the next chapters and may be dealt with when reaching Chapter 5.]

1.3.2A graphic method

Let us check it for the mass by using a graphic method which is particularly instructive. We are going to show that the properties of chirality of the theory forbid any mass counterterm for fermions (and by supersymmetry for bosons). The mass term for fermions is written

6We are supposing here that the integrals are properly regularized and thus that there exists a regularization procedure that respects supersymmetry.

Supersymmetry as a solution to the problem of naturalness 15

The convention is that all arrows are incoming or outgoing at the vertex. The scalar propagator (associated with the contraction φ φ) is denoted:

Indeed, we can define a direction on the propagator of the scalar field, a direction conserved by interactions, because we may associate it by supersymmetry with the chirality of its fermionic partner.

At the one-loop level, the self-energy of fermions is given by the diagram:

Let us represent chiralities in the case corresponding to a mass counterterm:

In either diagram, the two vertices impose contradictory directions for the arrow of the scalar propagator. Hence, we cannot write a contribution at one loop: there is no mass counterterm for fermions. This is a direct consequence of the chiral nature of the interactions.

Let us note that:

(i) One can easily write a contribution for wave function renormalization

¯ ¯

ΨL i/∂ΨL + ΨR i/∂ΨR :

16The problems of the Standard Model

(ii)the argument was given in terms of the complex field φ because this field is reminiscent, in its interactions, of the chirality of the spinor field associated. We could have argued in terms of the real fields A and B:

A B

The i2 = −1 in the pseudoscalar contribution plays the rˆole of the minus sign in the fermion loop, which is central in the cancellation of quadratic divergences. We check

here the importance of the fact that supersymmetry involves a complex scalar field

√

((A + iB)/ 2).

1.3.3Soft breaking of supersymmetry

Nonrenormalization theorems are characteristic of global supersymmetry. But nature is obviously nonsupersymmetric and if supersymmetry be, it must be broken. However, for what concerns us here – the cancellation of quadratic divergences, supersymmetry is not strictly speaking necessary.

Let us imagine for example that, in the Wess–Zumino model, we modify the mass squared of scalar A by δm2A or B by δm2B :

This is obviously not compatible with supersymmetry, as can be seen from the mass spectrum. The scalar contributions become

Supersymmetry as a solution to the problem of naturalness 17

and thus the cancellation of quadratic divergences (but not of logarithmic divergences) follows without problem. One represents generically these terms breaking supersymmetry by

Let us note that δM 2 destroys the symmetry between A and B (hence as well the cancellations between A and B just described).

On the other hand, a mass term for fermions

generates quadratic divergences:

A

δ m

The reason is that, in a supersymmetric theory, there is a cancellation between the following tadpole diagrams (see Exercise 2):

This cancellation is destroyed by the mass term above. It might appear surprising that one may shift scalar but not fermion masses. But one should not forget that it is the relation between fermion masses and other couplings (for example of A(A2 + B2)) in (1.35) which is responsible for the supersymmetric cancellation of quadratic divergences.

Another possibility is a modification of the couplings of the form:

δ = φ3 + φ 3 = A A3 3AB2 , (1.39)

LSB A √ −

2

18 The problems of the Standard Model

known as the A-term. Indeed such a term provides only cubic scalar interactions which cannot lead to quadratically divergent two-point functions. And the combination between A3 and AB2 is such that the contributions to tadpole diagrams (A) and (B) above cancel. On the other hand, a term such as

would lead to quadratic divergences.

The only other possibility lies in the gauge sector, if present: it is a mass term for the gauginos, supersymmetric partners of the gauge fields,

As we will see in Chapter 5, one can show that mass terms for scalars and pseudoscalars, A-terms, and mass terms for gauginos represent the only possible terms breaking supersymmetry without generating quadratic divergences. One expresses this property by saying that, under these conditions, supersymmetry is softly broken. The terms just discussed are the only ones which may be induced through a soft breaking of supersymmetry. A mass term for fermions corresponds on the other hand to a hard breaking of supersymmetry.

Further reading

•G. ’t Hooft in Recent Developments in Gauge Theories, Carg`ese 1979, NATO ASI series B, Vol. 59, Plenum Press, p. 135–157.

Exercises

Exercise 1 Majorana and Dirac masses.

The purpose of this first exercise is to familiarize the reader with the notions of Majorana, Weyl, and Dirac spinor. This is done here on the example of the neutrino, as in Section 1.1.1, but will prove to be useful in the rest of the book when we deal with supersymmetric particles.

Consider a spinor Ψ. One defines its chirality eigenstates

Ψc ≡ CΨT

where C satisfies CγµT C−1 = −γµ. In the representation chosen, CT = C† = −C and

CC† = C†C = 1.

(a) Show that

(Ψc)L (x) = C(ΨR )T (x),

(Ψc) (x) = C(Ψ )T (x).

R L