Elsevier

Annals of Physics

Volume 351, December 2014, Pages 751-772
Annals of Physics

Guises and disguises of quadratic divergences

https://doi.org/10.1016/j.aop.2014.10.002Get rights and content

Abstract

In this contribution, we present a new perspective on the control of quadratic divergences in quantum field theory, in general, and in the Higgs naturalness problem, in particular. Our discussion is essentially based on an approach where UV divergences are parameterized, after being reduced to basic divergent integrals (BDI) in one internal momentum, as functions of a cutoff and a renormalization group scale λ. We illustrate our proposal with well-known examples, such as the gluon vacuum self energy of QCD and the Higgs decay in two photons within this approach. We also discuss frameworks in effective low-energy QCD models, where quadratic divergences are indeed fundamental.

Introduction

The Higgs boson discovery at the LHC  [1](mH125GeV) as well as the lack of data supporting on low energy extensions to the standard model (SM) such as supersymmetry (SuSy) has renewed the interest in possible explanations for both the electroweak hierarchy and the naturalness problem. These issues are related to a certain extent to how we interpret quadratic divergences in field theories. Two kinds of hierarchy problems arise in the SM  [2]. The most commonly referred one, which we will simply call hierarchy problem, is related to the large radiative corrections to the Higgs mass stemming from quadratic divergences in the cutoff which supposedly cancel against the tree level value to a very high precision at the weak scale. Consequently the Higgs mass becomes quadratically sensitive to a cutoff scale Λ. On the other hand the gauge hierarchy problem has to do with logarithmic divergences which determine the running of the coupling constants: if on one level the scale that characterizes the symmetry breaking of the GUT which unifies quantum chromodynamics and electroweak theory  [3] is 1014  GeV, on the other level the electroweak symmetry breaking scale is about 102  GeV. Explaining this gap is known as the gauge hierarchy problem  [4]. The solution of the hierarchy problem involves how one bypasses the quadratic divergences which, unlike other divergences of a renormalizable theory which are multiplicatively renormalized, lead to a subtractive renormalization of the Higgs boson mass. Thus the hierarchy problem is reduced to the naturalness of such subtraction. SuSy avoids such subtractive renormalization and would solve the technical naturalness of the SM should SuSy particles be sufficiently light.

It is important to remark that contrarily to the interpretation in ultraviolet (UV) complete theories, quadratic divergences cannot be excused away as an artifact of the regularization procedure by simply adopting dimensional regularization, for instance. Taking the SM as the low energy limit of a more complete theory, a cutoff must be introduced to set a landmark in which new degrees of freedom appear. Notice that the meaning of a cutoff Λ is twofold: it can play the role of the UV cutoff of an UV complete theory (Λ) or a cutoff in an effective theory at which new degrees of freedom appear (merging scale). For example, for low energy models of QCD, Λ1GeV, as quarks and gluons are not well defined degrees of freedom in this region  [5]. Evidently, for both the ultraviolet complete and the effective theory, naive subtraction of quadratic divergences has no effect upon the low energy dynamics. However in drawing conclusions about new physics, such subtraction becomes a subtle and relevant issue.

However, as pointed out in  [6], the absence of quadratic divergences does not fully solve the hierarchy problem. The SM must also be UV completed at the scale Λ by a theory without quadratic divergences. Thus the problem also passes by at which scale a complete theory (say, SuSy) appears. That is because a matching of the parameters of the high energy and low energy physics ought to guarantee light Higgs mass parameters at the merging scale. Such fine tuning would be avoided only if Λ103GeV   [6].

Different constructions, differing by their level of sophistication, have appeared in the literature to explain away the role played by quadratic divergences in the naturalness problem, given that new physics has not been found at LHC with s=8TeV. It is worthwhile to discuss some proposals to give a panorama on the subject. For instance, naturalness without SuSy was proposed and studied by Jack and Jones in  [7], whereas in  [8] it was constructed a non-SuSy hypothetical theory which has the same particle content as softly broken minimal supersymmetric QED. It was shown that such theory was gauge invariant and free of quadratic divergences up to two loop order.

The oldest and widely discussed proposal is the Veltman condition  [9], by which in the SM CV=32mW2+34mZ2+34mH2fnfmf2,nf=3 for quarks and nf=1 for leptons, would make the coefficient of the Λ2 contribution to mH2 vanish if CV=0. However, this leads to mH=316GeV.

In  [10], within the scotogenic model of neutrino masses, the introduction of two scalar doublets, distinguished by Z2 symmetry, leads to two Veltman conditions which, in principle, could satisfy the vanishing of quadratic divergences without narrowing the Higgs mass as much. An alternative to the Veltman condition would be the compositeness of the Higgs particle by a strong infrared dynamics, forming a fermionic bound state, which at high energy breaks into its elementary fermionic constituents and, hence, quadratic divergences would be absent  [11].

On the grounds that neither the Veltman condition is satisfied for the measured value of the Higgs mass (electroweak scale) nor SuSy has been found at the LHC energies, Ref. [6] supposed that Veltman condition could be satisfied at some large energy μV where SuSy dominates (see also  [12]). Whereas simply imposing CV=0 leads to mH=316GeV, at odds with the current value, in terms of physical masses and couplings (1) is supposed to be renormalization group invariant. Thus Veltman condition (1) at one loop order can be written in terms of running couplings as  [6]CVħ(μ)=6λ(μ)+94g2(μ)+34g2(μ)6yt2(μ), where μ is the renormalization scale, λ is the Higgs potential self coupling, yt is the top Yukawa coupling and g,g are the electroweak gauge couplings. Setting CVħ(μ)=0 allows us to infer at which scale the Veltman condition is fulfilled (higher loop order corrections leads only to a small modification in μVħ   [13], [14]). A NNLO calculation for the running couplings  [6], using as inputs mH=126GeV,mt̄(mt)=161.5GeV and α3(mZ)=0.1196 for the strong coupling constant at the Z boson mass, leads to CVħ(μ)=0 at μ slightly larger than the Planck scale, which means that the SM is fine tuned up to this scale. Another adjustment in the parameters of the high energy fundamental theory must be performed in order to keep the Higgs and other singlets masses light at the merging scale. Such fine tuning is however unrelated to quadratic divergences. The appealing feature of this construction is that it puts off the solution of the hierarchy problem to the high energy complete UV theory.

In  [15] it was introduced new degrees of freedom through adding other contributions to Higgs boson wave function renormalization. Effectively, those new degrees of freedom change the Higgs boson coupling and, guided by naturalness, the authors construct a weak scale effective theory in which the new extra scalar fields cancel the quadratic divergences. They also argue that the parameter space of their “natural theories” can be tested to percent level precision through Higgs boson coupling measurements at LHC.

It is noteworthy that there have been claims which establish the Higgs lightness due to huge cancellations because an anthropic selection destroyed naturalness  [16]. Along similar lines, some authors claim a finite naturalness scenario in the sense that quadratic divergences are simply put aside (ignoring uncomputable power divergences) so that the Higgs mass is naturally small at least until there are no heavier particles  [17], [18]. They verify that finite naturalness is satisfied by the SM whilst for its extensions it remains valid only if the new physics is not much above the weak scale.

An interesting analysis on naturalness of the SM and extensions based on Bayesian statistics was performed in  [19]. ATLAS and CMS  [1], [20] operates in 20/fb with center of mass energy in the range s=7–8 TeV and will continue searching for SuSy to 13 TeV. Moreover a s=100TeV Very Large Hadron Collider (VLHC) may be constructed  [21]. Roughly speaking, Bayesian statistics is a numerical estimate of belief in a proposition (model), given the experimental data. Such an estimation is weighed by the Bayes-factor B. Unsurprisingly the evidence for the SM without quadratic divergences over SM with quadratic divergences, given both the mZ and mH measures, is huge (B1030, given that B=150 is considered very strong in the Jeffrey’s scale  [19]). A comparison between the likelihood of the SM and the constrained minimal supersymmetric SM (CMSSM)  [22] indicated, using as inputs the measured values of mH,mZ and LHC at 20/fb, that the Bayes factor favors the CMSSM over the SM with quadratic divergences by ≈1030, whereas SM without quadratic divergences is favored over the CMSSM by ≈700. Before the LHC measurements, this factor would be only ≈2. This is related to the “fine tuning price”. They conclude their paper arguing that natural models are most probable and naturalness is not simply an aesthetic principle. Moreover, the fine tuning price of null results from the VLHC (≈400) would be slightly less than that of LHC (≈500).

In this contribution, we point out another perspective on the control of quadratic divergences in quantum field theory, in general, and in the Higgs naturalness problem, in particular. Our viewpoint is consonant with the works of Fujikawa  [23] and Aoki and Iso  [2], but justifies them at a prior regularized level led by symmetry constraints.

We illustrate our proposal with well-known examples such as the gluon vacuum self energy of QCD and the Higgs decay in two photons within this approach. We also discuss frameworks in effective low-energy QCD models, where quadratic divergences are indeed fundamental.

Our discussion is essentially based on an approach where UV divergences are parameterized, after being reduced to basic divergent integrals (BDI) in one internal momentum, as functions of a cutoff and a renormalization group scale λ   [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. This construction, which was called Implicit Regularization (IR), can be generalized to arbitrary loop order to define the leading divergence of a Feynman diagram after subtraction of sub-divergences, as dictated by the local version of the BPHZ forest formula, based on the subtraction of local counter-terms  [41], [42], [43], [44], [45]. Thus, it complies with locality, Lorentz invariance and unitarity. The BDI’s can be absorbed in the definition of renormalization constants without being explicitly evaluated. This defines a minimal subtraction scheme, where the BDI’s are the bare bones of a Feynman amplitude UV behavior. The derivatives of BDI’s with respect to an arbitrary mass scale, say λ2, are also expressible in terms of BDI’s, which, in turn, have a lower superficial degree of divergence. Therefore renormalization group functions can be consistently evaluated within this approach.

In order to address the hierarchy problem, for instance, it is necessary to introduce a cutoff. The relations involving derivatives of BDI’s mentioned above are regularization independent and must be satisfied by any explicit regularization. Thus, we can use such relations to build a general parameterization for the BDI’s in terms of Λ. Contact with other explicit regularizations such as Pauli–Villars, dimensional regularization (DReg), sharp cutoff, Proper Time, etc. turns out to be immediate.

Arbitrary regularization dependent terms, which may be responsible for symmetry breaking in the underlying model, will be systematically displayed as surface terms (ST). Such ST’s can be systematically derived at arbitrary loop order, being defined as specific differences between BDI’s with the same superficial degree of divergence and different Lorentz structure, as we explain in the next section. The general parameterization we construct for each BDI clearly displays the ST undetermined character, which is fixed by symmetry requirements. That is because each BDI itself contains undetermined and regularization dependent parameters. In this case, usually they can be hidden in the arbitrariness of defining a renormalization constant. However, as we shall see in the case of quadratic divergences in the hierarchy problem, they may break symmetries as well.

This work is about arbitrary regularization dependent parameters, more specifically in the case of quadratic divergences, and how symmetry constraints which fix such parameters shed light on issues such as the hierarchy problem. In the latter, the scaling argument of Bardeen  [46] and conformal anomaly can be used to fix a undetermined parameter in the isolated quadratic divergence which contribute to the Higgs boson mass. A generalization of this strategy to higher loops is presented. This strategy follows Jackiw proposal in  [47] by which undetermined regularization dependent parameters must be fixed via symmetry and/or phenomenology constraints. In this way, we show that his strategy is accomplished not only to finite models, in which finite quantum corrections cannot be excused away by renormalization group conditions, but also to renormalizable and effective models, notably in studying quantum symmetry breaking.

To gain insight on how we deal with arbitrary parameters in a regularization independent way, as well as to give a general overview on quadratic divergences in QFT, we discuss the appearance of quadratic divergences in QCD, in the electroweak Higgs decay in two photons and in the quarkonium light meson decay in two photons. Finally we comment on the importance of quadratic divergences in low-energy QCD effective models. We organized the presentation as follows: in Section  2, we present an overview of the Implicit Regularization approach, showing how regularization dependent terms can be consistently identified; in Section  3, we show in the context of QCD how quadratic divergences naturally cancel themselves for the gluon self-energy; in Section  4, we discuss the Higgs decay to two photons, showing that the arbitrariness present in such case, although at first glance has a quadratic origin, is connected only with gauge symmetry; in Section  5; we discuss the role played by quadratic divergences in the gap equation of Effective Field Theories of QCD; the hierarchy problem in our formalism is presented in Section  6, in which we show how it is related to the ambiguities coming from quadratic divergences; we conclude in Section  7.

Section snippets

Basic divergent integrals, regularization ambiguities and parameterizations

In this section, we carry out a review of Implicit Regularization, as well as we discuss regularization ambiguities and general parameterizations of regularization dependent quantities. Within the approach of Implicit Regularization, the original divergent integral is assumed to be implicitly regularized (see  [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]). This allows algebraic manipulations in the integrand. To isolate the basic loop

Example: cancellation of quadratic divergences and renormalization of QCD at one loop

In this section, we show that quadratic divergences that appear in gluon self energies cancel out as they should, since they organize themselves into quadratic surface terms which are set to zero on gauge invariance grounds. We take the opportunity to evaluate the beta function of QCD using a different approach from the one presented in  [33]. In the present case, we will show, relying on the parameterization of BDI’s just presented in the last section, how a cutoff can be introduced while

Higgs decay to two photons

In this section we will discuss how to fix arbitrariness involved in the calculation of the Higgs decay to two photons. This decay was a subject of discussion in the recent literature ([37] and references therein). We follow the framework presented in Section  2. We will consider only the W boson loop, since it already contains all relevant aspects regarding arbitrariness we intend to discuss in the following. As showed in  [37], the diagrams in the unitary gauge that contribute are shown in

Quadratic divergences and effective theories: Nambu–Jona-Lasinio model

Contrarily to the previous examples, quadratic divergences play a vital role in the description of dynamical chiral symmetry breaking in models of low energy QCD, such as the Nambu–Jona-Lasinio model  [55]. This model belongs to the class of non-renormalizable Lagrangians and the regulator, usually expressible in terms of a cutoff Λ for the UV divergent one-loop quark integrals appearing at leading order of Nc, is characteristic of the scale at which spontaneous breakdown of chiral symmetry

Hierarchy problem

As discussed in the introduction, many proposals have been devised to interpret the role and fate of quadratic divergences in the hierarchy problem. It is necessary to introduce a cutoff to serve as a merging scale when we study the SM as an effective theory. A general parameterization for ultraviolet divergences with an explicit scale Λ much greater than the characteristic masses of the model can be constructed, as we have demonstrated in earlier sections. The parameterization of the quadratic

Concluding remarks

In this paper a discussion was carried out on the role of quadratic divergences in quantum field theory. This discussion was based in a general parameterization of basic divergent integrals. These basic divergent integrals, obtained in the context of Implicit Regularization, contain all the divergent content of a given amplitude. The parameterization we adopted embodies possible results coming from different regularization procedures. Arbitrary constants which naturally appear in the procedure

Acknowledgments

M.S., A.R.V and A.L.C. thank CNPq and FAPEMIG for financial support. M.S. thanks Durham University for the kind hospitality. This work has been partially supported by the Fundação para a Ciência e Tecnologia, the iniciative QREN, financed by UE/FEDER through COMPETE—Programa Operacional Factores de Competitividade. This research is part of the EU Research Infrastructure Integrating Activity Study of Strongly Interacting Matter (HadronPhysics3) under the 7th Framework Programme of EU, Grant

References (66)

  • E. Gildener

    Phys. Rev. D

    (1976)
  • I. Masina et al.

    Phys. Rev. D

    (2013)
  • M.T.M. van Kessel

    Nucl. Phys. B

    (2008)
  • M. Veltman

    Acta Phys. Polon. B

    (1981)
  • E. Ma

    Phys. Rev. D

    (2006)
    E. Ma,...
  • N. Craig et al.

    Phys. Rev. Lett.

    (2013)
  • G. Gazzola et al.

    J. Phys. G

    (2012)
  • K. Hepp

    Comm. Math. Phys.

    (1966)
  • K.G. Wilson et al.

    Phys. Rep.

    (1974)
    J. Polschinski

    Nuclear Phys. B

    (1984)
  • M.K. Volkov et al.

    Sov. J. Nucl. Phys.

    (1985)
  • I. Antoniadis et al.
  • Tech. Rep. ATLAS-CONF-2013-034

    (2013)

    Tech. Rep. CMS-PAS-HIG-13-005

    (2013)
  • H. Aoki et al.

    Phys. Rev. D

    (2012)
  • P. Langacker

    Phys. Rep.

    (1981)
  • M. Harada et al.

    Phys. Rep.

    (2003)
  • I. Jack et al.

    Phys. Lett. B

    (1990)
    I. Jack et al.

    Nuclear Phys. B

    (1990)
  • M.A. Zubkov

    Phys. Rev. D

    (2014)
  • M. Chaichian et al.

    Phys. Lett. B

    (1995)
  • M.S. Al-Sarhi et al.

    Z. Phys. C

    (1992)
  • Y. Hamada et al.

    Phys. Rev. D

    (2013)
  • V. Agrawal et al.

    Phys. Rev. Lett.

    (1998)
  • M. Farina et al.

    J. High Energy Phys.

    (2013)
  • Gouvea

    Phys. Rev. D

    (2014)
  • A. Fowlie, Supersymmetry, Naturalness and the Fine Tune Price at the VLHC,...
  • S. Chatrchyan, et al. (CMS Collaboration), 2014,...
  • Future Circular Collider Kickoff Meeting, Université de Genève, Genève,...
  • S.F. King et al.

    Phys. Rev. D

    (1995)
    O. Buchmueller

    Eur. Phys. J. C

    (2012)
    M. Kadastik

    J. High Energy Phys.

    (2012)
  • K. Fujikawa

    Phys. Rev. D

    (2011)
  • L.C. Ferreira et al.

    Phys. Rev. D

    (2012)
  • O.A. Battistel et al.

    Phys. Rev.

    (1999)
  • A.P. Baeta Scarpelli et al.

    Phys. Rev. D

    (2001)
  • A.P. Baeta Scarpelli et al.

    Phys. Rev. D

    (2001)
  • M. Sampaio et al.

    Phys. Rev.

    (2002)
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