{'fraction_non_alphanumeric': 0.03924468206119895, 'fraction_numerical': 0.013422818791946308, 'mean_word_length': 4.296385542168674, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Horava-Lifshitz Type Quantum Field Theory and Hierarchy Problem

We study the Lifshitz type extension of the standard model (SM) at UV, with dynamical critical exponent z=3. One loop radiative corrections to the Higgs mass in such a model is calculated. Our result shows that, the Hierarchy problem, which has initiated many excellent extension of the minimal SM, can be weakened in the z=3 Lifshitz type quantum field theory.


I. INTRODUCTION
Our current description of the basic interactions in nature, based on the standard model (SM) of particle physics and general relativity, is in spectacular agreement with all known experiments. However, it is almost certainly fundamentally incomplete. The extreme finetuning needed to keep the Higgs mass small compared to the Planck scale (i.e., the Hierarchy Problem) has motivated many extension of the minimal SM. All of these contain new physics, beyond the SM, which might be tested at the Large Hadron Collider (LHC). The most widely explored of these new physics is supersymmetry (SUSY).
Recently, a new quantum field theory of gravity with "dynamical critical exponent" z equal to 3 in the UV was introduced in [1]. Such a theory is power-counting renormalizable [2] in the 3+1 spacetime and can be quantized using stochastic quantization method [3].
Besides, it exhibits scaling properties which are anisotropic between space and time Measuring dimensions of operators in the unit of spatial momentum, we get [x] = −1 and [t] = −z. This system doesn't process the Lorentz invariance for nontrivial exponent z (z = 1) but processes spatial rotational and translational invariance. The prototype of such a quantum field theory with z = 1 is the theory of a Lifshitz scalar in D+1 dimensions, first proposed as a description of tricritical phenomena. The action of the Lifshitz scalar can be written as where ∆ is the spatial Laplacian. This action describes a free-field fixed point with critical exponent z = 2. Such fixed points with anisotropic scale invariance are called Lifshitz points.
Properties of Lifshitz type field theory have been investigated in [4]. The construction of gauge theories with Lifshitz fixed points in D+1 dimensions have been discussed in [5,6].
Such a theory flows naturally to the relativistic value z = 1 at long distance, and therefore the Lorentz symmetry shall appear as an emergent symmetry.
In this paper we assume that the SM has a Lorentz non-invariant UV-completion and realize it at a Lifshitz fixed point with critical exponent z = 3. A similar extension of the SM in Lorentz violating approach is discussed in [7,8]. We focus on the one-loop radiative corrections to the Higgs mass. Our results show that the Hierarchy problem can be weakened in such a theory.
The outline of the paper is as follows: In section II, we will construct the SM at the Lifshitz fixed point. In section III, we will calculate the one-loop radiative corrections to the Higgs mass. Some conclusions are drawn in Section IV.

II. THE MODEL
In this section we construct the SM at the Lifshitz fixed point and derive propagators for scalar, vector and spinor fields, which will be used for the loop calculation in the next section. We first write down the interactions for the spinor and Higgs fields, with dynamical critical exponent z, which can be written as where ψ represents the SU(2) L doublet or singlet, γ i (i = 0 ∼ 3) is the gamma matrix and is the covariant derivative. The engineering dimensions of fields ψ and H, as well as coupling constants α and β, are given by The system has a free-field fixed point with z = 3 for any spatial dimensions. The Yukawa interactions between the Higgs and spinor fields are Here [Y α ] = 3/2(z − 1). A novel approach responsible for the origin of fermion mass in a Lifshitz type extension of the SM including an extra scalar field has been proposed in [9].
In this paper, we simply assume that Yukawa interactions are responsible for the origin of fermion masses after electroweak symmetry spontaneously broken.
Now we switch to constructing the Yang-Mills theory with arbitrary dynamical critical exponent z in 3+1 dimensions. We take A a 0 , A a i as the time and spatial components of gauge fields, separately. Our theory should be invariant under the following gauge transformation: Gauge invariant Lagrangian will be constructed from the field strengths, which are constructed from the commutations of covariant derivatives as [D t , D i ] = igE i and [D i , D j ] = igF ij . The Lagrangian should contain a kinetic term which is quadratic in first time derivatives. Following the strategy proposed in Ref. [6], we obtain the following gauge invariant where Tr represents the trace for the gauge generators and δ denotes dimensionless coupling constant. The engineering dimensions of the gauge field components and coupling constants at the corresponding fixed point z are The equation of motion for the gauge fields can be obtained from Eq. (8) by varying A 0 : Similar to Ref. [6], we can choose the following natural gauge-fixing condition A 0 = 0 , and According to the equation of motion, once we adopt the gauge-fixing condition in Eq. (10) at t = t 0 , this condition will continue to hold for all t. Then the propagator for gauge fields can be obtained by using functional method We can also derive the propagators for spinor and Higgs fields from Eq. (3) and (4), using the same method: Actually, interactions given in Eq. (3), (4) and (8) etc. That is why we use the symbol " ∝ " instead of " = " in Eqs. (11) and (12). However, when Λ > Λ IR , where Λ IR stands for infrared cut off, below which the Lorentz symmetry is recovered, these terms will be the subdominant contribution to the propagators. As a result, We can safely use these propagators to preform one-loop calculations at scale Λ(> Λ IR ). A distinctive feature of the Lifshitz type quantum field theory is that the dispersion relation [10], where α i are marginal coupling parameters.

III. HIERARCHY PROBLEM
For a long time, there were only two solutions to the Hierarchy Problem: SUSY and technicolor, and SUSY is heavily favored. In recent years, there are several other new ways to address the Hierarchy problem, including ADD models [11], little Higgs models [12], twin Higgs models [13], folded SUSY [14], Lee-Wick SM [15], and so on. In this section, we will explore a new solution to the Hierarchy problem in the Lifshitz type quantum field theory.
We will assume that z = 3 and calculate one-loop radiative corrections to the Higgs mass in such a theory, using the propagators for scalar, spinor and vector fields presented in Eqs. (11) and (12). Relevant feynman diagrams are listed in Fig. (1).
where λ is four Higgs self-interaction coupling constant and Λ UV is ultraviolet cut off. We can find that there is only logarithmic divergence instead of quadratic divergence in Eq.
(13). Fig. 1 (b) comes from Yukawa interactions. Direct calculation to this diagram results in where To calculate integral in Eq. (14), we have used the approximation p ≪ k, which is good enough for p ≪ Λ IR . Fig. 1 (c) comes from gauge interactions of Higgs field, which gives where p is the spatial momentum of Higgs field and g 1 , g 2 are gauge coupling constants corresponding to U(1) Y and SU(2) L , respectively.
To sum up, traditional feynman diagrams listed in Fig. 1, that are quadratic divergent in the SM, become logarithmic divergent in the z = 3 Lifshitz type quantum field theory.
Actually, in this theory there are two other feynman diagrams that contribute to Higgs mass.
We list them in Fig. 2. Direct calculation to Fig. 2 (a) results in where Q ≡ 1/4(6g 4 2 + 3g 2 1 g 2 2 + g 4 1 ). while calculation to Fig. 2 where R ≡ 1/8(10g 6 + 6g 2 1 g 4 2 + 3g 4 1 g 2 2 + g 6 1 ). It is clear from Eqs. (16) and (17)  There also exist some other terms that contain Higgs field in our theory, such as κ gf ℓ C L g εHℓ T L f εH, which is renormalizable and could be responsible for the origin of small Majorana neutrino masses. But this term doesn't contribute to Higgs mass at one-loop level.
In summary, the Hierarchy problem is weakened in our model.

IV. SUMMARY
In this paper we have considered a Lifshitz type extension of the SM at UV with dynamical critical exponent z = 3. We have written down the full interactions and derived the propagators for scalar, spinor and vector fields. Then we have focused on calculating oneloop radiative corrections to the Higgs mass. Our results show that,the Hierarchy problem can be avoided or weakened in the z = 3 Lifshitz type quantum field theory. But still, there are many other problems for such a theory, which are important and interesting but beyond the scope of this paper. A detailed study to these topics will be shown in somewhere else.