Lorentz Violation from the Higgs Portal

We study bounds and signatures of models where the Higgs doublet has an inhomo- geneous mass or vacuum expectation value, being coupled to a hidden sector that breaks Lorentz invariance. This physics is best described by a low-energy effective Lagrangian in which the Higgs speed-of-light is smaller than c; such effect is naturally small because it is suppressed by four powers of the inhomogeneity scale. The Lorentz violation in the Higgs sector is communicated at tree level to fermions (via Yukawa interactions) and to massive gauge bosons, although the most important effect comes from one-loop dia- grams for photons and from two-loop diagrams for fermions. We calculate these effects by deriving the renormalization-group equations for the speed-of-light of the Standard Model particles. An interesting feature is that the strong coupling dynamically makes the speed-of-light equal for all colored particles.


Introduction
The sector responsible for the electroweak symmetry breaking still leaves open theoretical questions and is experimentally unknown. Its most plausible explanation relies on the idea of spontaneously broken gauge symmetry, although the Higgs mechanism introduces its own problems. The main puzzle is associated with the presence of a mass parameter for the Higgs field, which sets the scale for the electroweak phenomena. At the quantum level this mass term is quadratically sensitive to short-distance physics. Actually, being the only superrenormalizable interaction in the Standard Model, this mass term can be viewed as a window open towards the influence of new and unknown high-energy or hidden sectors of the theory. New scalars M (x), neutral under the SM gauge group, can have renormalizable couplings to the Higgs H: This aspect was discussed by several authors and was dubbed "Higgs portal" in [1]. In this paper we consider the possibility that this Higgs portal connects the Standard Model with some hypothetical sector that breaks Lorentz invariance, such that M (x) has a space-time dependent vacuum expectation value (vev) varying on a characteristic small length-scale .
Violation of Lorentz invariance is not uncommon in certain theories of quantum gravity, as in the presence of a space-time foam, and even in string theory. Alternatively the Higgs field itself might be 'foamy', existing only in tiny islands of space-time. Or maybe its vev might be 'foamy', being non-zero only in some regions, giving rise to a small average vev from a larger fundamental vev. In both cases an apparently constant Higgs vev is obtained at low energy, i.e. after averaging over length-scales much bigger than . Here we study the low energy signals of these kinds of scenarios, performing concrete computations from the interaction in eq. (1).
In section 2 we compute the Lorentz non-invarant dispersion relation satisfied by a scalar or a fermion with a non-constant mass M (x). In section 3 we develop a general technique to obtain the full effective Lagrangian. In section 4 we write RGE equations for the speed-of-light of the various SM particles, finding how Lorentz-breaking in the Higgs sector propagates at loop level to all other particles, and how a strong coupling can dynamically restore the Lorentz symmetry. In section 5 we consider the signals and constraints, and in section 6 we conclude.

Propagation of particles with space dependent masses
One of the consequences of the scenario we consider is a non-constant Higgs vev, and consequently a non-constant mass for SM particles. In order to obtain some physical intuition about our setting, we start by considering the propagation of a complex scalar particle or of a Dirac fermion with masses that vary periodically in space.

Scalar
We study the case of a complex scalar H(x) with a squared mass M 2 (x) that depends only on one spatial coordinate x. We assume that M 2 (x) has period , and constant values M 2 1 and M 2 2 within intervals of length r 1 and r 2 (with 0 < r 1,2 < 1 and r 1 + r 2 = 1): According to the Floquet-Bloch theorem [2], as a result of the periodicity of M 2 (x), the solution of the Klein-Gordon equation is of the form where u(x) also has periodicity . The Klein-Gordon equation for u(x) is given by and has the solution Continuity of the function u(x) and of its first derivative at the matching points x = 0 and x = r 1 imposes four constraints. Three of them determine the integration constants A 1,2 and B 1,2 up to an overall normalization, while the fourth equation defines the dispersion relation: This equation describes the relation between energy E and momentum k. We see that k is fixed up to a 2π/ ambiguity.
Since we are assuming that Lorentz violation is related to phenomena at very short distance, we are interested in particle propagation for momenta much smaller than 1/ . In this limit, the dispersion relation in eq. (7) can be written in the familiar form where Therefore, when the particle is observed at momenta much smaller than 1/ , the effect of a space-varying mass can be absorbed in a redefinition of its mass and of its "light speed" (or, more appropriately, of the maximal attainable velocity in the massless limit). While the mass redefinition is unobservable (unless we have a theory in which particle masses can be predicted), the redefinition of the "light speed" can be experimentally measured when the particle propagation is compared with another particle with different value of c. Thus, a scalar particle with a space-varying mass, when viewed at low energies, appears as a particle with a constant mass, given by the square root of the average of M 2 (x), but with a modified relation between energy and momentum.
The Lorentz-violating effect trivially disappears when M 2 1 → M 2 2 , since the source of Lorentz violation vanishes in this limit. More interestingly, Lorentz violation also disappears when → 0. In this limit, the characteristic length of the mass variation becomes infinitely smaller than the de Broglie wavelength of the particle. Since the source of Lorentz violation M 2 (x) has dimension of mass squared, the adimensional correction to c must be suppressed by the high scale Λ = 2π/ . Thereby, in our scenario, high-scale physics generates small Lorentzbreaking effects. This is unlike a generic Lorentz-breaking scenario (such as 'space-time foam'), where one typically expects order unity deviations from c = 1 even from Lorentz-violation at the Planck scale. 1 The correction to c in eq. (9) is always negative and thus the "light speed" of a scalar is smaller than the canonical value.

Fermion
We can now repeat the discussion in the case of a fermion. Suppose that its Dirac mass depends on x with period , being constant within intervals of lengths r 1 and r 2 Again, we are considering only one space dimension. Exploiting the Floquet-Bloch theorem, we can decompose the fermion field as where the 2-component spinors u (±) are periodic functions. In the Weyl basis, the Dirac equation becomes The solution is where k 1,2 are again given by eq. (6). The conditions of continuity of the functions u (±) at the matching points x = 0 and x = r 1 determine, up to their normalization, the integration constants A 1,2 and B 1,2 and the dispersion relation In the limit of small momenta, the energy-momentum relation is given by the familiar expres- Again, at small momenta, the total effect of the space-varying mass can be parametrized by a distortion of the "light speed". Notice that the "light speed" of the scalar deviates from the canonical value at order 4 while, for space-varying fermion masses, the effect comes already at order 2 . This is because the Lorentz-violating effect is always suppressed by two powers of the mass inhomogeneity, which amounts to (M 2 1 − M 2 2 ) 2 for the scalar and (M 1 − M 2 ) 2 for the fermion. Dimensional arguments then determine the different powers of . As in the case of the scalar, the correction to c in eq. (15) is negative and thus the maximal attainable velocity is smaller than the ordinary light speed.

Low-energy effective theory
After having clarified the physical meaning of a particle with space-varying mass, we can proceed in the analysis of the Standard Model with a Lorentz-violating Higgs mass parameter. Our goal is to construct an effective theory valid at energies below a cutoff scale Λ, obtained by integrating out the high-frequency modes. Here 1/Λ represents the typical length of the variations of the Higgs mass. Since Λ is the energy scale at which the Lorentz violation, originating in a hidden sector, is communicated to the Higgs field, we assume that Λ is much larger than the TeV scale. The space dependent mass M (x) mixes the low-frequency modes (with Fourier momentum k Λ) with the high-frequency ones. By integrating out the highfrequency modes, their effects is described at low energy by Lorentz-violating operators. Let us explain the procedure to derive the effective theory.

Higgs effective Lagrangian
We denote with H the Higgs doublet and introduce in the Lagrangian a space-time dependent component for its mass, Here M is a mass parameter of the order of the electroweak scale, µ is a mass parameter (much smaller than the electroweak scale) parametrizing the amplitude of the space-time varying component, and F is a generic order unity dimensionless function that modulates the space-time dependence. For simplicity we take F to depend on a single combination of where a is a fixed 4-vector. It is convenient to normalizex with −a 2 because we have in mind a space-like fluctuation of the Higgs mass (a 2 < 0), but our results remain valid also for time-like variations (a 2 > 0). We assume that F is a real function with the following three properties. (i) It is periodic: It averages to zero within one period: 2π 0 dxF (x) = 0. Being periodic, the function F can be expanded in an infinite sum of Fourier modes, The Fourier coefficients f n are such that f * n = f −n , because F is real, and such that f 0 = 0, because of the property (iii) above. Moreover, the f n are real (imaginary) when F is an even (odd) function ofx.
It is convenient to work in Fourier space and express the Higgs field H(x) as We can decompose the quadri-momentum k in terms of the quantities which have been defined such that k ⊥ · a = 0 and k 2 = k 2 ⊥ − k 2 . Note that k 2 is positive (negative) for space-like (time-like) fluctuations of the Higgs mass. The integration over k of a generic function g(k) can be decomposed into an infinite sum of integrations within shells of momenta (n − 1/2)/|a| < k < (n + 1/2)/|a| for any arbitrary integer n: Using this expansion, the Higgs action in eq. (16) becomes where the dependence of H on k ⊥ is understood.
The first line of eq. (22) is just the usual SM Lagrangian, that gives the ordinary Lorentzinvariant dispersion relation for each mode with momentum k. The term proportional to µ 2 in the second line of eq. (22) introduces a mixing between the different modes of the Higgs field. In particular, the zero mode of the Higgs field (n = 0) mixes with every high-frequency mode n with a coefficient µ 2 f n . Notice that the term proportional to µ 2 generates only off-diagonal mixings, since f 0 = 0. A non-periodic M 2 (x) would give a continuous (rather than discrete) mixing, but the final result would be the same as long as the Fourier transform of M 2 (x) vanishes fast enough at k → 0, so that an unambiguous splitting between low-momentum and high-momentum modes still exists.
Coming back to the periodic M 2 (x), the low-energy effective theory is obtained by integrating out all modes of the Higgs field H with n = 0. This procedure leads to a non-trivial result because of the mixing of the high-frequency modes with the zero mode. To obtain the effective theory, it is convenient to express the high-frequency modes through their equations of motion at first order in µ 2 , Replacing eq. (23) into eq. (22), retaining only terms involving zero modes (n = 0) and expanding the result for small |a|, we obtain the low-energy effective theory for the Higgs field: In eq. (24), the integration is only over momenta k smaller than the cutoff Λ = 1/|a|.
While Z and ∆M 2 can be absorbed in the wave-function and mass definitions, δc leads to a physical effect. Note that, for space-like variations of the Higgs mass (a 2 < 0), ∆M 2 is negative and one could imagine scenarios in which the electroweak breaking is triggered solely by Lorentz-violating effects.
In coordinate space, the new effect is described by a Lorentz-violating term: This is a renormalizable interaction. Its coefficient δc ∼ µ 4 /Λ 4 is however suppressed by four powers of the cutoff scale, at which the Lorentz violation is communicated to the Higgs sector. This operator modifies the Higgs kinetic term in such a way that the "light speed" for the Higgs field along the direction identified by the quadri-vector a becomes c = 1 + δc.
In this case f n = i[(−1) n − 1]/(πn) and thus we obtain Notice that this expression of δc coincides with the result in eq. (9) obtained by solving the Klein-Gordon equation with variable mass, after the replacement r 1,2 = 1/2, = 2π/Λ and |M 2 1 − M 2 2 | = 2µ 2 . So far we have studied the case in which the Lorentz violation identifies one special direction in space-time, but our results can be easily generalized. Actually the derivation of the effective theory used a generic 4-vector a and can be adapted to different cases. For instance, taking a 4-vector a with vanishing space components corresponds to the rotationally invariant case, which leads to a Lorentz-violating Lagrangian term When M 2 (x) has the symmetry of a cube (corresponding to the octahedral group), the dimension-4 effective operator is still of the form of eq. (31), exhibiting rotational symmetry. This is because the usual δ ij is the only two-index invariant tensor of both the octahedral group and the full SO(3) rotation group. The breaking of the rotational symmetry will appear only in higher dimensional operators. Another case leading to eq. (31) is the one in which M 2 (x) is a randomly-varying function. This gives the rotationally invariant effective operator, just like the random motion of molecules gives, on average, a rotationally invariant refraction index of air. In the following, just for simplicity, we focus on the rotationallysymmetric case, which contains all the important features of Higgs-induced Lorentz violation. Finally note that, after taking into account gauge corrections, the operator in eq. (31) gets gauge-covariantized as usual, ∇ → D = ∇ + ig A.

Full effective Lagrangian
The construction of the low-energy effective Lagrangian for the Higgs field can now be extended to the full Standard Model. Each Standard Model field is expanded in Fourier space and the high-frequency modes are integrated out. This procedure can be carried out with the help of the equations of motion, as discussed above, or, more simply, with the Feynman diagram technique. The propagator of the n-mode Higgs field, expanded for small |a|, is given by  Figure 1: The diagrams generating the effective Lorentz-violating interactions, obtained after integrating out the high-frequency modes, for the terms with two Higgs bosons (a) and two fermions (b). Single lines denote propagators of low-frequency modes for the Higgs boson (dashed line) and the fermion (solid line). Double lines denote propagators of the highfrequency modes, as given by eq. (32) for the Higgs boson and by eq. (34) for the fermion. The dot denotes the mixing between high-and low-frequency modes, as given by eq. (33).
The mixing between the zero-mode and any n-mode of the Higgs field corresponds to a mass insertion iµ 2 f n .
A summation +∞ n=−∞ is required in diagrams with high-frequency modes in the internal lines. Using these rules, we can easily recover the Lagrangian in eq. (24) from the Feynman diagram in fig. 1a. We can then extend the calculation to the other Standard Model fields. The Lorentz violation, originally residing in the Higgs mass term, is communicated to quarks and leptons through the tree-level diagram of fig. 1b. The propagator of the n-mode fermion field, expanded for small |a|, is given by where m is the mass of the fermion field. Hence, we obtain that the diagram in fig. 1b generates the following effective operator connecting two Higgs (H) and two fermion fields (ψ), Here λ is the corresponding Yukawa coupling. If ψ is a weak singlet, contractions of SU(2) L indices give |H| 2 ; if ψ is a weak doublet, ψ H is the component of ψ that gets mass from the Yukawa λ.
In the special cases in which the modulating function F (x) is a cosine or a square wave, .
The operator in eq. (35), after electroweak symmetry breaking, gives a modification of the "light speed" of the fermion along the direction identified by a: Here v = H is the Higgs vacuum expectation value. In sect. 2 we found that a spacedependent fermion mass gives a distortion to the "light speed" already at order a 2 . Instead, here we are finding that, if the Lorentz violation originates in the Higgs mass, the effect for the fermions starts only at order a 6 . The reason is that, in such a case the inhomogeneity in v and consequently in the fermion mass m = λv, is itself suppressed by two powers of a, due to the effect of the Higgs kinetic term. From an effective theory point of view, the Higgs vacuum expectation value is constant and does not break Lorentz invariance. Therefore fermions can feel the effect of Lorentz violation only through higher dimensional operators, like the one in eq. (35), induced by the mixing between the zero mode and the high-frequency modes of the Higgs field.
At tree level, the Lorentz violation is communicated to gauge fields through diagrams involving Higgs and gauge particles. These diagrams have the effect of making the derivatives contained in eq. (26) covariant under the gauge group and also generate some new higherdimensional operators.
In summary, we have considered the effects of a space-time varying Higgs mass in the Standard Model. The source of Lorentz violation is expressed in terms of two parameters: µ 2 , which characterizes the amplitude of the mass square variations, and |a| (or 1/Λ), which defines the wavelength (or frequency) of these variations. The most appropriate language to address the problem is that of an effective low-energy field theory, valid below the scale Λ. In the effective theory, all the effects are induced by the mixing of the Higgs modes, which is proportional to µ 2 /Λ 2 . The Higgs vacuum expectation value and all masses are constant in the effective theory, but the Higgs kinetic term is modified by a Lorentz-violating operator. This operator is renormalizable, but its coefficient is proportional to µ 4 /Λ 4 , and thus suppressed by four powers of the cutoff scale. All other effects can be written in terms of higher-dimensional operators, suppressed by additional powers of Λ. For instance, the Lorentz-violating effects in the fermion kinetic term are proportional to µ 4 v 2 /Λ 6 . These conclusions are based on tree-level considerations. Now we turn to discuss the effects of quantum corrections.

Lorentz violation at loop level
Let us consider the effective Lagrangian containing the dominant rotationally invariant dimension-4 Lorentz-violating operators, in the rest frame of the Lorentz-breaking sector 2 Here A = {Y, W a , G a } describes the SM gauge bosons, ψ are the 15 SM Weyl (chiral) fermion multiplets (L, E, Q, U, D, appearing in 3 generations) and H is the scalar Higgs. The coefficients δc are the corrections to their 'speed-of-light'. Each of the various δc can be set to 2 The most generic Lorentz-breaking Lagrangian in the notations of [3] can be reduced to this form setting (k φφ ) 00 = −2δc H , (k F ) i0i0 = (k F ) 0i0i = −(k F ) 0ii0 = −(k F ) i00i = δc A , (c ψ ) 00 = δc ψ . All other tensors vanish and all other components of these tensors vanish. In general, such tensors describe non-isotropic Lorentz violation [3]. Performing Lorentz transformations on eq. (38) the other components of the c ψ , k F , k φφ tensors are generated as dictated by their Lorentz structure.
that differs from the standard expression only because we specified the speed of light to be used in the various terms, e.g. p 2 = E 2 /c 2 − p 2 . The group theory coefficient is b H = 1/6 for both SU(2) L and U(1) Y (normalized such that the H hypercharge is 1/2). Thereby where m h is the physical Higgs mass and c γ = c 1 cos 2 θ W + c 2 sin 2 θ W for the photon. After a further loop correction, one also gets a δc for the SM fermions ( fig. 2b):

RGE for the speed-of-light
The loop effects are best described by a system of RGE for the speed-of-light of the various SM particles, that allows us as usual to re-sum the log-enhanced corrections.
We consider a generic theory with particles p (gauge vectors A, Weyl fermions ψ and scalars H) interacting among them with gauge couplings g A , Yukawa couplings λ. The quartic scalar couplings do not enter in the RGE under consideration. We find that the RGE equations for their maximal speeds are: 3 where b p are the well known coefficients that enter in the RGE for the gauge couplings: The group factors are defined as Tr T a T b = T 2 δ ab (T 2 = 1/2 for the fundamental of SU(n), and T 2 = n for the adjoint) and as (T a A T a A ) ij = C A δ ij (C 1 = q 2 for a U(1) charge; C 2 = 3/4 for the fundamental of SU(2); C 3 = 4/3 for the fundamental of SU (3)). The self-renormalization terms in eqs. (42) come from wave-function renormalizations. Note that the Lorentz-invariant limit c p = c is RGE invariant.
In the Standard Model, summing over the three generations of E, L, U, D, Q and using the GUT normalization g 1 = 5/3g Y , the RGE are: We neglect here the effect of Yukawa couplings. After the breaking of the electroweak symmetry, the SM fermions acquire Dirac masses m. In the non-relativistic limit, the speed-oflight c L and c R of the left and right handed components become the fermion speed-of-light c = (c L + c R )/2 plus a Lorentz-breaking Hamiltonian operator ∼ (c L − c R ) p · σ. At first order in perturbation theory around the Lorentz-symmetric state, p · σ changes the energy of one given state by an amount proportional to its matrix element, which vanishes being odd in p.

Signals and bounds
In sect. 3.1 we have found that Lorentz violation from the Higgs portal predicts, at tree level and at the RGE scale Λ ∼ 1/a, a negative correction to the Higgs speed-of-light, δc H ≈ −µ 4 /Λ 4 . Fig. 3 shows how the original Lorentz violation feeds into the various SM particles through the RG evolution from Λ = 10 12 GeV to the weak scale and then down to the QCD scale Λ QCD .
We note an interesting generic feature of the RGE: in the limit in which a gauge coupling becomes strong, all values of c of particles charged under the gauge group become equal. In particular, when the QCD coupling becomes strong, g 3 → ∞ at µ ∼ Λ QCD , all colored particles reach a common c: speed differences get exponentially suppressed by exp (−k g 2 3 d ln µ) factors, where k is a numerical constant. To compute the common c, one notices that the strong coupling does not renormalize the combination 16c 3 +9 q c q (summed over light quarks q). This means that Lorentz invariance can be dynamically emergent if all SM particles felt at some energy a strong coupling. This could be possible in a SU(5) model such that the unified coupling runs to a large enough value.  • Proton vacuumČerenkov radiation, namely p → pγ decays, would become kinematically allowed at E p > m p / √ c p − c γ [6]. Protons have been observed in cosmic rays up to E p ∼ 10 7 TeV, and thus c p − c γ < 0.9 × 10 −15 [6,7]. This bound is plotted as a dashed line: the change in its slope arises because we only consider cosmic ray energies below the cut-off Λ of our theory.
• Similarly, electron vacuumČerenkov radiation, namely e → eγ decays, would become kinematically allowed at E e > m e / √ c e − c γ , but cosmic ray electrons have been observed up to 2 TeV, so that c e − c γ < 10 −13 [8].
• Furthermore, the agreement of electron synchrotron radiation at the LEP accelerator with its standard expression implies |c e − c γ | < 5 × 10 −15 [10]. A numerically similar bound can be deduced from astrophysical observations of Inverse Compton and synchrotron radiation [11].
• Stability of various types of spectral lines despite the motion of the earth implies strong bounds on Lorentz-violating operators [12], but not on the δc operators present in our scenario (also considered in [6]), at leading order in δc. Subdominant bounds are listed in ref. [13]. 4 The picture also shows two more plausible values for the space-time dependent part µ of the Higgs mass: a) µ at the weak scale; b) µ such that its contribution to the Higgs mass in the low-energy effective theory ∆M 2 ∼ µ 4 2 (negative for space-like inhomogeneities) of eq. (25) is at the weak scale. In such a case electroweak symmetry breaking could be a byproduct of inhomogeneities; the result ∆M 2 µ 2 holds because inhomogeneities in the Higgs vev are suppressed by the Higgs kinetic term |∂ µ H| 2 ∼ v 2 / 2 .
We also considered sub-leading effects, suppressed by powers of E/Λ, which lead to variations in the speed-of-light that depend on the energy E. The main experimental constraints on such effect are: Such bounds are not competitive. Furthermore, in our scenario (and actually more in general) also the dominant effects depends on energy, due to the logarithmic RGE running of c. Since |δc H | |δc γ | in our model, c γ has a sizable RGE running above the weak scale, d ln c γ /d ln µ ∼ 10 −3 , and a much slower running at lower energies below Higgs decoupling, d ln c γ /d ln µ ∼ 10 −5 . The limits in eq. (45) thereby imply a bound |δc γ | < 10 −12 , which is again not competitive with the constraints previously described.
The main qualitative point is that Lorentz violation in the Higgs sector must be suppressed by a scale well above the electroweak scale. This means that various possible solutions to the Higgs mass hierarchy problem that one can invent using Lorentz violation (e.g. assuming that the Higgs is a 2d field localized on strings that fill the space; or adding spatial gradients | ∇H| 4 to the Lagrangian) are experimentally too strongly constrained to make the weak scale naturally small. portal: only the Higgs mass term µ 2 |H| 2 violates Lorentz invariance, being inhomogeneous on small scales 1/Λ. This naturally leads to a small correction to the Higgs speed-of-light, δc H ∼ −(µ/Λ) 4 . We computed this effect at tree level in two ways: i) in section 2 we have solved the propagation equations in a simple inhomogenous background; ii) in section 3 we have derived an effective Lagrangian: inhomogeneities lead to mixing between low and high-frequency modes, so that the integration out of the high-frequency modes gives a Lorentzviolating effective operator, |( ∇ + ig A)H| 2 . Fermions are affected only by higher dimension operators, which we have computed.
At loop level, δc H propagates to all other SM particles via a system of RGE equations for their speed-of-light; fig. 3 shows a typical solution. An interesting feature is that the strong coupling dynamically drives the speed-of-light of all colored particles to a common value. The signals and bounds of our scheme of Lorentz violation were explored in section 4.