Electroweak Symmetry Breaking without the $\mu^2$ Term

We demonstrate that from a low energy perspective a viable breaking of the electroweak symmetry, as present in nature, can be achieved without the (negative sign) $\mu^2$ mass term in the Higgs potential, thereby avoiding completely the appearance of relevant operators, featuring coefficients with a positive mass dimension, in the theory. We show that such a setup is self consistent and not ruled out by Higgs physics. In particular, we point out that it is the lightness of the Higgs boson that allows for the electroweak symmetry to be broken dynamically via operators of $D\geq 4$, consistent with the power expansion. Beyond that, we entertain how this scenario might even be preferred phenomenologically compared to the ordinary mechanism of electroweak symmetry breaking, as realized in the Standard Model, and argue that it can be fully tested at the LHC. In an appendix, we classify UV completions that could lead to such a setup, considering also the option of generating all scales dynamically.


Introduction
A common lore, in particular after the discovery of the Higgs boson, is that electroweak symmetry breaking (EWSB) is triggered by a negative sign mass term for a Higgs doublet, either introduced by hand or generated dynamically. The ultraviolet (UV) completion of the Standard Model (SM) is expected to generate (approximately) a Higgs potential of such a form. However, as we will entertain in this article, a more fundamental theory of nature could also have an opposite low energy limit, where the appearance of relevant operators is fully avoided. As we will show, in such a scenario an 'irrelevant' D = 6 operator of the type O 6 = |H| 6 could induce a non-trivial vacuum for the scalar sector. 1 We will point out that it is the lightness of the Higgs boson that allows to consider this special setup, where the operators in the Higgs potential are not only deformed in a sub-leading way, as a phenomenologically viable alternative to the SM-form of the potential. It will turn out that this lightness, m 2 h v 2 , makes it consistent with the effective field theory (EFT) expansion to fully trade the (D = 2) µ 2 -operator for O 6 .
After having demonstrated the self-consistency of the setup, including the stability with respect to quantum corrections within the EFT, we will turn to the phenomenology of the model. The most important collider observable is Higgs-boson pair production, where we will show that the LHC is capable of fully testing the pure version of the idea. Beyond that, the model has intriguing consequences for cosmology. We will see that it just lies in the correct ballpark such as to allow for a strong first order phase transition, as required by electroweak baryogenesis. While in the main part of the paper we just treat the µ = 0 Higgs potential as a distinct and interesting boundary condition that a potential UV model could fulfill, thereby opening up a new direction in model building, in the appendix we will present and classify possible ideas for UV completions of our setup.

The Form of the Higgs Potential
We consider the SM without relevant operators and instead augment the Higgs potential with a dimension-6 term c 6 /Λ 2 O 6 such that it takes the simple form All dimensionfull parameters are either zero or at the cutoff of the theory. Inspecting the form of the potential, a first observation is that a stable and nontrivial minimum at |H| 2 > 0 should be possible if λ < 0 and c 6 > 0. In the following, we will check if such a minimum is also viable phenomenologically. For any given cutoff scale Λ, we can first calculate the position of the minimum, i.e., the vacuum expectation value (vev), denoted as Clearly, the minimization condition only fixes the relative size of the coefficients λ and c 6 /Λ 2 . 2 In turn, an electroweak-scale vev can be obtained without the need for a large coefficient of the D = 6 operator O 6 . The size of the latter will however get fixed by the mass of the physical Higgs excitation around the vev, h, where in 3) The consequences of these relations will be scrutinized in the next section.

Self Consistency of the Setup
We will now examine quantitatively, if it is possible to generate the vev and the Higgs mass at the correct values, keeping the parameters in the range of the validity of the EFT, and then discuss the stability of the low energy setup. To that extent, we first solve eqs. (2.2) and (2.3) for the two free parameters in the potential, λ and c 6 , expressing them in terms of the vev, fixed by the Fermi constant as v = 246 GeV, and the Higgs mass m h ≈ 125 GeV. We obtain We inspect that, since m 2 h /v 2 ≈ 1/4 1, a large cutoff Λ 2 v 2 is possible while keeping c 6 ∼ O(1). We can thus see explicitly that it is the lightness of the Higgs boson which allows for the mechanism to work. The required c 6 versus the cutoff Λ is visualized in Figure 1, where the perturbativity limit c 6 < 4π is given by the horizontal dashed line. In particular, setting Λ = 0.8 TeV (Λ = 1 TeV) requires only c 6 = 1.8 (c 6 = 2.8) while even Λ = 2 TeV is still possible in a rather strongly coupled setup with c 6 = 11.4 < 4π. On the other hand, perturbativity of the underlying theory suggests that around Λ ≥ 2 TeV, at the latest, new physics (NP) should be found. 3 If the new states are uncolored (which we will assume in the following), such mass scales clearly introduce no tension with current LHC limits. Moreover, we have checked that for all values of the cutoff considered above, the inclusion of a D = 8 operator with O(1) coefficient alters the numerical results by only a few per cent or less. It is thus save to neglect such operators.
We will now study more detailed the correlation between the needed size of the coefficient c 6 /Λ 2 and the physical parameters in the Higgs sector, m h and v, stressing that only a limited part of the larger parameter space, considered before the discovery of the Higgs boson, is viable in our model. To finally remove the redundancy of parameters in the low energy theory, allowing for an easier discussion, we redefine thereby eliminating the explicit reference to the new scale Λ (which we shall leave open).
In the left panel of Figure 2 we depict the required value ofc 6 versus the Higgsboson mass. The red dashed linec 6 ≡ 1 roughly corresponds to the upper boundary of the viable parameter space, not in conflict either with perturbativity or the validity of the EFT. Values above that line would require c 6 4π for Λ 850 GeV. We can see clearly that a heavy Higgs boson of only m h 300 GeV would have already basically invalidated our approach. The same would have been true for a smaller vev of v < 100 GeV (keeping m h = 125 GeV), which we show for completeness in the right panel of Figure 2. The experimental values m h = 125 GeV and v = 246 GeV, visualized by green vertical lines, are however in perfect agreement with a reasonable value ofc 6 ≈ 0.17. Finally, the potential (2.1), employing these values, is plotted in the left panel of Figure 3. It exposes the expected mexican-hat form, featuring a stable minimum at a non-trivial field value. We conclude that, while it would have been easily possible that the numerical values of the mass scales generated in nature after EWSB would have excluded our setup, the actual values just lie in a range that allows for EWSB to be triggered by a single D = 6 operator instead of a negative mass squared term.
In order for the strict µ 2 = 0 setup to be self-consistent, the µ 2 term should also not be generated again from loop effects within the effective theory, involving physical scales. This is however guaranteed, since the only physical suppression scale Λ can always be factored out of loop integrals and never enters dynamically.
Finally, we show that the inclusion of the SM quantum corrections to the potential, generating a term of the form |H| 4 log(H 2 /µ 2 r ) [1], only corresponds to a small perturbation of our setup, leading to little changes in the parameters λ and c 6 . Neglecting the tiny impact of light quarks, the SM contributions to the one-loop Coleman-Weinberg potential are given by (see, e.g., [2]) Here, the tree-level field-dependent mass terms read where we have dropped contributions suppressed by Λ 2 , the numbers of degrees of freedom are n W = 6, n Z = 3, n h = 1, n χ = 3, n t = −12, (3.5) and the constants C i are given by In the end, the top quark furnishes the dominant correction. Adding (3.3) to (2.1), setting the renormalization scale to µ r = v/ √ 2, and solving for c 6 and λ that reproduce correctly v and m h , leads to the shifts which is a O(10%) effect. We show the resulting potential as a red dashed line in the right panel of Figure 3. It becomes a little bit flatter before the zero of the undisturbed potential and a bit steeper afterwards. Moreover, there arises a tiny maximum at low values of |H|, such that the origin is a minimum -however it corresponds to (V + ∆V ) = 0 and thus lies much higher than the global minimum at |H| = v/ √ 2. The overall changes are very modest. The solid line depicts the required value of the Wilson coefficient c 6 in dependence on the cutoff Λ in our setup, while the blue region allows for an appropriate first order electroweak phase transition such as to trigger electroweak baryogenesis. In the right plot we focus on the region with lower Λ. See text for details.

Phenomenology
Beyond the potential direct discovery of new states at scales of (1−2) TeV, our model offers distinct signatures in Higgs-pair production and cosmology that we want to discuss in the following.
First of all, the sizable coefficientc 6 ≈ 0.2 leads to a notable change in the production cross section of Higgs Pairs, since O 6 contributes to the trilinear Higgs-self interaction after EWSB. In fact it increases the cross section by a factor of ∼ (2 − 3). This is in a range that should be possible to exclude at the LHC with a luminosity of L 600 fb −1 , see [3]. 4 Beyond that, the presence of the operator O 6 also modifies the electroweak phase transition. Without this operator, the phase transition is of second order for m h = 125 GeV (see, e.g., [2]). This excludes the possibility of electroweak baryogenesis within the SM as there is no out-of equilibrium dynamics at the phase transition. On the other hand, a sizable contribution of O 6 changes the Higgs potential such that a first order phase transition becomes possible for the physical Higgs mass [4], allowing for electroweak baryogenesis (if enough CP violation is present). In Figure 4, we show again c 6 versus the cutoff Λ, where now the blue region corresponds to a first order phase transition that leads to a stable T = 0 minimum, while in addition sphaleron processes are sufficiently suppressed in the broken phase such as to not wash out the generated baryon asymmetry [4]. The latter requirement leads to the condition h(T c ) /T c We conclude that the required value ofc 6 leads to a very interesting phenomenology, allowing for pronounced effects in Higgs-pair production as well as opening the possibility of the creation of our current universe via electroweak baryogenesis. This makes the setup avoiding relevant operators attractive on its own. Beyond that, it calls for an examination of how the effective potential (2.1) could be generatedapproximately or exactly -from a UV theory. This will be discussed in Appendix A.

Conclusions
As we know very little about the dynamics of EWSB or how the HP is eventually solved in nature, various approaches to EWSB should be examined and tested, in particular also from the low energy perspective, even if they might not be the most obvious ones. In this article, we have demonstrated that setting the notorious relevant operator |H| 2 in the Higgs potential to zero and adding instead an operator O 6 = |H| 6 can lead to a viable electroweak symmetry breaking. We pointed out that it is the lightness of the Higgs-boson, m h = 125 GeV < v, that -perhaps unexpectedly -allows for this setup to be self-consistent, in a sense that it is in agreement with perturbativity and a NP scale of Λ ∼ (1 − 2) TeV. The latter is needed in order to create a mass gap that allows to neglect operators with D = 8 and beyond. Moreover, once set to zero, the µ 2 term is not re-generated in the low energy theory. Eliminating this parameter and adding instead the D = 6 coefficient c 6 keeps the theory very predictive, since the number of parameters stays the same. In particular, the setup is fully testable in experiments currently under way, since relatively large changes in the Higgs-pair production cross section are predicted.
The main purpose of our analysis was to study the phenomenological viability of this distinct IR limit, allowing new directions in model building. However, in the appendix we will also provide already a first classification of UV completions that could lead to such a setup, at least approximately, or exactly, depending on further structures of the UV theory. There, we will also entertain the possibility that, while the Higgs boson could be a fundamental scalar, all scales are generated dynamically.
As it is a distinct theoretical limit, which also opens the possibility of generating our universe via baryogenesis at the electroweak scale and interestingly enough is not ruled out by measurements in the Higgs sector yet, the µ 2 → 0 model examined here should be considered as an alternative mechanism of breaking electroweak symmetry dynamically.

A Possible UV Completions
So far, the form of the Higgs potential (2.1) was just a matching condition on the unspecified UV completion. Now, we are going to discuss classes of UV models that could generate such a potential at the tree level. The picture will always be that the SM-like theory (featuring µ 2 = 0) possesses no scale at the classical level and is then coupled to a sector that breaks scale invariance, potentially dynamically. Note that such an additional breaking is needed in the first place, since it was found that the breaking of scale invariance within the SM by dimensional transmutation is not sufficient to generate the Higgs mass of m h = 125 GeV (see, e.g., [5,6]). The NP might itself respect scale invariance at the classical level, generating all masses dynamically. However in the first part of the following discussion we will leave open how the scales in the new sector emerge.
Let us nevertheless already stress a difference of our approach from the usual one, often used in models employing classical scale invariance (CSI) as a building principle. In the latter, the µ 2 term is forbidden at the tree level, but then regenerated spontaneously, usually via the (loop-induced) vev of an additional scalar singlet in a Higgs portal term, mimicking the usual SM Higgs potential (see, e.g., [5,6], as well as in general [7] on models that generate all scales dynamically). Thus, the low energy phenomenology can be quite similar to the one of the SM. In our approach, however, no relevant operator needs to be generated in the electroweak-scale theory at all. The breaking of scale invariance is induced in an orthogonal -possibly also spontaneous/dynamical -way, via an irrelevant operator, introduced by integrating out a heavy field that couples to the SM. This leads to a distinct low energy phenomenology and full testability of our setup. It provides a new minimal way of allowing for a viable EWSB in the presence of the scale-invariant tree-level SM Lagrangian, that interestingly features m h → 0 in the decoupling limit of Λ → ∞. 5

A.1 General Discussion
We will start with the option of not worrying weather CSI is a good building principle for the full Lagrangian. Depending on how the NP scale is generated, the full models might possess CSI or not. We will focus on three minimal realizations, where in each we introduce a new field transforming in a different representation of SU (2) L .
Fourplet of SU (2) L A scalar field Φ transforming as a fourplet of SU (2) L has the intriguing feature of being able to directly generate the operator O 6 at the tree level, allowing the NP to be heavy, while a potential contribution to the |H| 2 operator can be deferred to the multi loop level (or beyond). Here, we use the fact that three Higgs doublets H can combine to a weak fourplet. The relevant interaction of Φ with the Higgs Boson takes the schematic form λ 3 (ΦH 3 ) 0 , where the subscript 0 denotes the full singlet combination. We neglect portal terms, quadratic in both H and Φ, which could be motivated via compositeness of Φ, leading to D[Φ 2 ] > 2, or just considered as an simplifying assumption. We assume a TeV scale mass term M 2 Φ |Φ| 2 such as to generate O 6 at the correct scale Λ ∼ M Φ ∼ TeV. Interestingly, no direct mixing between the Higgs doublet and the new scalar Φ is allowed and, although the latter has week gauge interactions, since it is not colored, it should be save from current limits. A survey of the impact of the new fields from the UV completion on Higgs phenomenology (and precision observables) is left for future work. The detailed impact will depend on the concrete realization of the UV theory, such as its quantum numbers, scalar potential, and masses, but can be rather modest, given the large mass of the new fields not stemming from the SM Higgs sector.
In this setup, the operator O 6 is generated via tree level exchange of the field Φ, as depicted by the leftmost Feynman diagram in Figure 5, leading to c 6 ∼ |λ 3 | 2 . 6 A contribution to µ 2 on the other hand can be avoided at both the tree-and oneloop orders -it only arises potentially at two loops by closing four external Higgs legs, realizing already approximately (2.1). The latter contribution could further be canceled by invoking (partial) supersymmetry, which leads to vanishing quadratic quantum corrections to µ 2 , while still the tree-level generation of O 6 is possible in general. A related discussion on a complete cancellation of UV effects on the |H| 2 operator -which however there is generated again spontaneously -is given in [6] (see also [5]). 7 Alternatively, one could imagine the λ 3 interaction terms to be cut off by a strong form factor, such that λ 3 (p) is rapidly vanishing for large momentum p flowing through the vertex. While this would tame loops, it would again not prevent a sizable tree level contribution to O 6 via integrating out Φ at zero momentum.
Singlet of SU (2) L Generating O 6 at the right scale by integrating out a scalar singlet S, with mass M S , is somewhat more involved. The singlet can in general feature two interaction structures with the Higgs boson, M S λ S S|H| 2 and λ p S 2 |H| 2 , and both are needed to induce O 6 at the tree level, as shown in the second diagram in Figure 5, leading to c 6 /Λ 2 ∼ λ p |λ S | 2 /M 2 S . Note in that context that generating c 6 at the loop level would be borderline, since in the region consistent with our assumptions (perturbativity, validity of the EFT expansion) it would in general tend to be be too small. If one wants to insist on generating all scales dynamically, the dimensionfull coefficient in front of λ S could be thought of as a vev of a new field, see below. Without additional structures, this UV model features however less natural separation between the size ofc 6 and contributions to renormalizable operators (µ 2 would be generated at one loop) -thus additional ingredients such as supersymmetry or other UV cancellations, together with some adjustment of parameters, seem to be needed more strongly than in the case of the fourplet.
Doublet of SU (2) L Finally, a doublet ϕ of SU (2) L with a TeV scale mass can also generate the operator O 6 at the tree level via an interaction term λ ϕ 3 |H| 2 ϕ † H, similar to the fourplet. The corresponding diagram is shown in the rightmost panel of Figure 5. Compared to the fourplet however, again more assumptions (like CSI, compositeness/supersymmetry) are needed to suppress a potential mixing between the scalar doublets as well as the impact of interaction terms with more powers of ϕ.
We conclude that, while several UV setups could lead to the potential (2.1), a fourplet of SU (2) L seems to offer the most straightforward candidate.

A.2 Dynamical Generation of all Scales
Now we are also going to address how the scale in the UV completions emerges, showing concrete possibilities to generate all scales dynamically. It will thus not be necessary to assume a relevant operator at all, neither in the shortest-distance UV theory, nor in the IR limit E ∼ v. We will first consider compositeness as a rather natural mechanism to obtain a mass from binding energy, without the need of a relevant operator in the UV theory. Then we will discuss the option of generating a mass from the Coleman-Weinberg mechanism.

A.2.1 Compositeness
One elegant option to obtain the mass for the particles to be integrated out is to consider them to be composites, bound together by a new strong interaction. We assume the condensation scale, beyond which one could resolve the fundamental constituents, to lie well above 1 TeV, such that at currently accessible energies one would never see them directly.
Fourplet of SU (2) L As a first option one can consider a set of two fermions, forming a massive scalar fourplet. The simplest options are a doublet and a triplet, a fourplet and a singlet, or a fourplet and a triplet. These fields could be massless before they condense -however the UV constituents could, depending on their quantum numbers, possibly also get a mass from the Higgs sector after EWSB. Moreover, the fourplet could stem from a condensation of two scalars, with the same quantum numbers as considered for the fermion case.
Singlet of SU (2) L In analogy to as discussed in the last paragraph, two fermions or scalars could also condense to form a massive singlet of SU (2) L , given that they transform both in the same representation. Beyond the mass of this composite state, as discussed above, one needs a scale to be present in the S|H| 2 interaction. This in turn would also need to be generated dynamically, like via the vev of a further singlet from the Coleman-Weinberg mechanism (see below).
Doublet of SU (2) L Finally, a set of fermions or scalars could bind together to create a doublet of SU (2) L . Here the simplest options are a doublet combining with a singlet or triplet as well as a triplet bound together with a fourplet.

A.2.2 Coleman-Weinberg Mechanism
Before concluding, let us also entertain the option to generate all scales via the Coleman-Weinberg mechanism. The simplest Coleman-Weinberg setups always rely on a non-negligible vev for a scalar in order to generate its mass [5,6]. To avoid potential issues with precision tests or Higgs physics from SU (2) L breaking (or, in the case of the SU (2) L singlet, the direct generation of the µ 2 term), in our case we need to assume the presence of one more scalar particle compared to the general discussion at the beginning of the section, which however could be elementary. In practice, another singlet S 2 could obtain a vev from the Coleman-Weinberg mechanism as explained before (see also [6]). In turn, portal interactions with the SU (2) L representations introduced in Section A.1 would generate their masses dynamically. On the other hand, direct portals to the SM Higgs need to be strongly suppressed in the thought setup. This could e.g. be realized geometrically in an extra-dimensional framework, with the multiplets to be integrated out being the messengers between the SM and the new singlet S 2 , residing on different branes.
Finally, nature might have chosen a completely different way to generate O 6 , while avoiding the µ 2 term, still to be found. A further analysis of the potential UV completions, including the examination of dark matter candidates, will be deferred to future work.