Higgs mass bounds from renormalization flow for a simple Yukawa model

Holger Gies, Clemens Gneiting, and René Sondenheimer

Phys. Rev. D 89, 045012 – Published 19 February 2014

Abstract

We study the functional renormalization group flow of a Higgs-Yukawa toy model mimicking the top-Higgs sector of the standard model. This approach allows for treating arbitrary bare couplings. For the class of standard bare potentials of $ϕ^{4}$ type at a given ultraviolet cutoff, we show that a finite infrared Higgs mass range emerges naturally from the renormalization group flow itself. Higgs masses outside the resulting bounds cannot be connected to any conceivable set of bare parameters in this standard model $ϕ^{4}$ class. By contrast, more general bare potentials allow us to diminish the lower bound considerably. We identify a simple renormalization group mechanism for this depletion of the lower bound. If this depletion is also active in the full standard model, Higgs masses smaller than the conventional infrared window do not necessarily require new physics at low scales or give rise to instability problems.

Received 12 September 2013

DOI:https://doi.org/10.1103/PhysRevD.89.045012

Authors & Affiliations

Holger Gies^1,*, Clemens Gneiting^2,†, and René Sondenheimer^1,‡

¹Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, D-07743 Jena, Germany
²Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany

^*holger.gies@uni-jena.de
^†clemens.gneiting@physik.uni-freiburg.de
^‡rene.sondenheimer@uni-jena.de

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Vol. 89, Iss. 4 — 15 February 2014

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Images

Figure 1
Extended mean-field analysis of the lower bound for the Higgs mass $m_{H}$ versus the UV cutoff $Λ$ , based on a bare potential $U_{Λ}$ of $ϕ^{4}$ type for $N_{f} = 1$ . For an initial UV potential which is flat apart from a mass term $U_{Λ} = \frac{1}{2} m_{Λ}^{2} ϕ^{2}$ , the fermionic fluctuations drive the Higgs mass to a finite minimal value. The solid lines correspond to standard mean-field theory accounting only for top fluctuations, cf. Eq. (25), whereas the dashed lines also include scalar fluctuations on the Gaussian level (extended mean field). The four different line sets correspond to increasing values of the initial $ϕ^{4}$ coupling of $λ_{Λ} = 0$ , $\frac{1}{6}$ , $\frac{1}{3}$ , $\frac{2}{3}$ from bottom to top.
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Figure 2
Extended mean-field analysis of the lower bound for the Higgs mass $m_{H}$ versus the UV cutoff $Λ$ based on a bare potential $U_{Λ}$ of $ϕ^{6}$ type for $N_{f} = 1$ . We have plotted the lower bound in the $ϕ^{4}$ theory ( $λ_{Λ} = λ_{3, Λ} = 0$ ) as the solid black line. Theories with bare couplings $λ_{Λ} = - \frac{1}{30}$ and $λ_{3, Λ} = \frac{2}{3}$ are depicted as the red dashed line, and $λ_{Λ} = - \frac{1}{15}$ and $λ_{3, Λ} = 2$ as the blue dotted line.
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Figure 3
Higgs mass values versus the bare scalar coupling $λ_{Λ}$ for a cutoff $Λ = 1 0^{7} GeV$ . The dashed lines denote the results within LO derivative expansion; the NLO deviates from the LO result by at most 10% for large coupling, demonstrating the satisfactory convergence of the derivative expansion. Also, the convergence of the polynomial expansion is shown: green lines with squares are obtained within the lowest nontrivial order with $N_{p} = 2$ ; blue lines with circles denote the $N_{p} = 4$ result; even higher orders $N_{P} = 6$ , 8 show no further deviation from the $N_{p} = 4$ curves.
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Figure 4
Higgs mass bounds versus cutoff $Λ$ . The thick green/solid line denotes the lower bound for the Higgs mass derived within the class of bare $ϕ^{4}$ potentials. The thin red/solid line shows the lower bound as derived within mean-field approximation. The turquoise/dashed lines mark upper bounds if the bare scalar coupling is allowed to start maximally from $λ_{Λ} = 1$ , 10, 50, 100 from bottom to top, respectively. An artificial restriction to the perturbative domain $λ_{Λ} ≲ 1$ underestimates the upper bound by a factor $≳ O (1)$ .
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Figure 5
Higgs mass versus cutoff $Λ$ . The black upper line denotes the lower bound for the Higgs mass derived within the class of bare $ϕ^{4}$ potentials. The red/gray lower line shows how we can construct Higgs masses below the lower bound by giving up the restriction to quartic bare potentials. The masses are derived for $λ_{2} = - 0.1$ and $λ_{3} = 3$ .
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Figure 6
Estimate for the radius of convergence in units of $1 0^{3} {GeV}^{2}$ of the polynomial expansion of the effective potentials in terms of the absolute values of the ratios of expansion coefficients. The red filled circles are derived for a theory which starts at $Λ = 1 0^{7} GeV$ with all couplings set to zero apart from the mass term. The black empty circles are for the case $Λ = 1 0^{7} GeV$ and $λ_{2} = 1$ and $λ_{n} = 0$ ( $n \geq 3$ ).
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