Fun with the Abelian Higgs model

Abstract

In calculations of the elementary scalar spectra of spontaneously broken gauge theories there are a number of subtleties which, though it is often unnecessary to deal with them in the order-of-magnitude type of calculations, have to be taken into account if fully consistent results are sought for. Within the “canonical” effective-potential approach these are, for instance: the need to handle infinite series of nested commutators of derivatives of field-dependent mass matrices, the need to cope with spurious IR divergences emerging in the consistent leading-order approximation and, in particular, the need to account for the fine interplay between the renormalization effects in the one- and two-point Green functions which, indeed, is essential for the proper stable vacuum identification and, thus, for the correct interpretation of the results. In this note we illustrate some of these issues in the realm of the minimal Abelian Higgs model and two of its simplest extensions including extra heavy scalars in the spectrum in attempt to exemplify the key aspects of the usual “hierarchy problem” lore in a very specific and simple setting. We emphasize that, regardless of the omnipresent polynomial cut-off dependence in the one-loop corrections to the scalar two-point function, the physical Higgs boson mass is always governed by the associated symmetry-breaking VEV and, as such, it is generally as UV-robust as all other VEV-driven masses in the theory.

Appendices

Appendix A: Handling the infinite series of nested commutators

Denoting for simplicity ${M}_{S}^{2}\equiv A$ and ${\partial {M}_{S}^{2}}/{\partial\phi_{a,b}}\equiv A_{a,b}$ the infinite series in Eq. (5) reads

(A.1)

where the commutator in the kth term is taken k−1 times. Interestingly, it is often the case that the {A,A _a } ×[A,..[A,A _b ]..] part of the kth term above (which we shall denote $f^{k}_{ab}$) can be written as a (k−1)th power of a certain matrix B commuting with A which is further multiplied from the left by a constant matrix pre-factor C, i.e.,

$$ f^{k}_{ab}=CB^{k-1}, \quad[A,B]=0. $$

(A.2)

Then, however, one can sum up the series (A.1) quite easily:

(A.3)

In the simplest case above, i.e., whenever one can implement (A.2), it is clear that C={A,A _a }A _b and $B=A_{b}^{-1}[A,A_{b}]$. Note that in the derivation (A.3) we also assumed that B was invertible. Actually, the invertibility of B is not really necessary as, in practice, the RHS of formula (A.3) can be defined by its limit even for a singular B. Finally, for B=0, the inner series in Eq. (5) reduces to just its first term and the formal limit

$$\lim_{B\to0}CB^{-1} \bigl[\log(A+B)-\log A \bigr]= CA^{-1} $$

is also retained.

Appendix B: The self-energies and the fate of the spurious IR divergences

In what follows we shall use the asymmetric-phase¹¹ Lagrangian (10) to calculate the scalar-sector contribution to the momentum-dependent part of the Higgs self-energy (which is all we need for the difference Σ _HH (p ²)−Σ _HH (0)). The relevant Feynman diagrams are

(B.1)

and it is clear that only the second graph develops a spurious IR divergence in the p ²→0, m ²→λv ² regime. The result of a simple calculation reads

(B.2)

where $I(p^{2},m_{1}^{2},m_{2}^{2})$ denotes the basic loop integral

(B.3)

and C_UV denotes the standard UV-divergent $\overline{\rm MS}$ structure in the dimensional regularization, i.e., $\mathrm {C}_{\mathrm {UV}}= \frac {1}{\varepsilon}-\gamma_{E}+\log4\pi$ in d=4−2ε. Hence, there is a term of the form −λ ² v ²/8π ²log[(−m ²+λv ²)/μ ²] in the finite shift Σ _HH (p ²)−Σ _HH (0) which compensates the spurious IR divergence in the zero-momentum-squared formula (21) when the Higgs pole mass is determined from Eq. (22).

Appendix C: The Higgs sector beta and gamma functions

In this appendix we shall focus namely on the determination of the λ-dependent parts of the anomalous dimension $\gamma _{H}^{(\lambda)}$ of the Higgs field and of its quartic coupling beta function $\beta_{\lambda}^{(\lambda)}$ in the minimal Abelian Higgs model discussed in Sect. 3.3; these are the key ingredients to formula (23).¹²

3.1 C.1 The λ-dependent part of the Higgs anomalous dimension

It is quite simple to see that there is no contribution to $\gamma _{H}^{(\lambda)}$ at the one-loop level. The reason is that the trilinear couplings in the relevant scalar “blob” diagrams (B.1) are not dimensionless and, thus, the loop integration does not generate a momentum-squared-dependent contribution to the UV divergence. As a consequence, δZ _H does not contain a term proportional to λ and, thus, at one loop, γ _H in a generic R _ξ gauge receives only contributions from the gauge(+Goldstone) sector so one concludes $\gamma_{H}^{(\lambda)}=0$.

3.2 C.2 The λ-dependent part of the Higgs quartic coupling beta function

For what follows it is convenient to write down the scalar part of the relevant Lagrangian in the broken phase including counterterms

(C.1)

where, for simplicity, we imposed the tree-level stationarity condition (which makes no difference here—we will be anyway interested only in the UV divergences). Given δZ _H and δλ the one-loop Higgs quartic coupling beta function can be written as

$$ \beta_{\lambda}= -\lambda\frac{\partial K_{\lambda}}{\partial\log\mu}+ 2\lambda\frac{\partial K_H}{\partial\log\mu}, $$

(C.2)

where K _λ =λ ⁻¹ δλ and K _H =δZ _H . Note that in dimensional regularization one has

$$ \frac{\partial K_{\lambda,H}}{\partial\log\mu}= -2\varepsilon K_{\lambda,H} + \text{higher order terms}, $$

(C.3)

so, indeed, it is sufficient to consider the UV pole structure of the relevant diagrams.

Since K _H at the one-loop level does not develop a purely λ-proportional UV divergent term (see Sect. C.1) for our purposes here it is sufficient to evaluate only the proper vertex renormalization factor K _λ . This is done by considering the diagrams of the type:

(C.4)

An elementary calculation yields the following contributions to the four-point function:

which are compensated by the counterterm $\Delta\varGamma_{\lambda^{2}}^{CT}= -6i{\delta\lambda}$ if and only if K _λ =5λ/8π ² ε+UV regular terms. Thus, one can conclude that the λ-dependent part of the one-loop Higgs quartic coupling beta function in the minimal Abelian Higgs model reads

$$ \beta_{\lambda}^{(\lambda)}=\frac{5}{4\pi^{2}}\lambda^{2}. $$

(C.5)

Appendix D: The second derivative of the effective potential and the pole mass

For the sake of completeness, let us just briefly recapitulate the derivation of the central formula (22) here; this material can be, of course, found in the existing literature, see, e.g., [14, 15] and references therein.

The physical (pole) mass is in any renormalization scheme S given by the root of the S-renormalized inverse propagator

$$ \varGamma_{S}^{(2)}\bigl(p^{2}\bigr)\equiv p^{2}-\mu_{S}^{2}-\varSigma _{S} \bigl(p^{2}\bigr)=0. $$

(D.1)

Using the fact that the second derivative of the (dimensionally regularized $\overline{\rm MS}$) effective potential (3) in the vacuum $\mathcal{M}^{2}$ corresponds to $\varGamma_{\overline{\rm MS}}^{(2)}(0)$ one has

$$ \mathcal{M}^{2}=-\mu_{\overline{\rm MS}}^{2}-\varSigma_{\overline{\rm MS}}(0) , $$

(D.2)

and, hence, the physical mass corresponds to the root of the equation

$$ p^{2}-\mathcal{M}^{2}-\varSigma_{\overline{\rm MS}} \bigl(p^{2}\bigr)+\varSigma _{\overline {\rm MS}}(0)=0. $$

(D.3)

This, however, is nothing but Eq. (22) up to a trivial generalization to matrices.