“Triviality” and the Perturbative Expansion in λΦ Theory

The “triviality” of (λΦ)4 quantum field theory means that the renormalized coupling λR vanishes for infinite cutoff. That result inherently conflicts with the usual perturbative approach, which begins by postulating a non-zero, cutoff-independent λR. We show how a “trivial” solution λR = 0 can be compatible with the known structure of perturbation theory to arbitrarily high orders, by a simple re-arrangement of the expansion. The “trivial” solution reproduces the result obtained by non-perturbative renormalization of the effective potential. The physical mass is finite, while the renormalized coupling strength vanishes: the two are not proportional. The classically scale-invariant λΦ4 theory coupled to the Standard Model predicts a 2.2 TeV Higgs, but does not imply strong interactions in the scalar sector. 1. Suppose we accept that the 4-dimensional λΦ4 theory is indeed “trivial” [1], meaning that it has no observable particle interactions; what is the theory’s effective potential? Since there are no interactions the effective potential can only be the classical potential plus the zero-point energy of the free-field fluctuations. This is the crucial insight of Ref. [2]:— for a “trivial” theory the one-loop effective potential is effectively exact. (A recent lattice calculation provides striking confirmation of this fact [3].) The usual perturbative renormalization [4] is then not appropriate because it would spoil this exactness — it does not properly absorb the infinities, but merely pushes them into “higher-order terms” which are then neglected. However, it is simple to renormalize the one-loop effective potential in an exact way [5, 6, 7, 2]. (This was first discovered in the context of the Gaussian effective potential [8, 9].) The constant background field φ, the argument of Veff , requires an infinite re-scaling, but the fluctuation field h(x) ≡ Φ(x)− φ (i.e., the pμ 6= 0 projection of the field) is not re-scaled [2]. The particle mass mh is related to the cutoff Λ and the bare coupling constant λ = λ(Λ) by mh = Λ 2 exp− 32π2 3λ . (1) Thus, for mh to remain finite λ must vanish like 1/ ln(Λ/mh) in the continuum limit (Λ → ∞). As a consequence one finds that the connected n-point functions at nonzero momentum vanish for n > 2, implying no particle interactions; i.e., “triviality”. In particular, the connected 4-point function, from which one might have hoped to define a renormalized coupling constant λR, vanishes. The usual perturbative approach, by contrast, is based on an attempt to generate a cutoff-independent and non-vanishing λR. No meaningful continuum limit is possible in perturbation theory. In fact, as discussed by Shirkov [10], perturbative calculations of the β function up to 5 loops [11] provide the following results: In odd orders, β pert , β pert , β 5−loop pert are positive and monotonically increasing. In even orders β 2−loop pert , β 4−loop pert each have an ultraviolet fixed point, which would imply a finite bare coupling constant, in contradiction with the rigorous results of Ref. [1]. The magnitude of this spurious fixed point at even orders appears to decrease to zero with increasing perturbative order. A Borel re-summation procedure [10, 11] yields a positive, monotonically increasing β function, as in odd orders. That does not allow a continuum limit because the renormalized coupling will have an unphysical Landau pole. The moral is that only by abandoning, at the start, the vain attempt to define a non-zero renormalized 4-point function can one obtain a continuum limit. In the effective potential analysis [2] one actually starts from an approximation scheme (one-loop or