Renormalization group theory of generalized multi-vertex sine-Gordon model

We investigate the renormalization group theory of generalized multi-vertex sine-Gordon model by employing the dimensional regularization method and also the Wilson renormalization group method. The vertex interaction is given by $\cos(k_j\cdot \phi)$ where $k_j$ ($j=1,2,\cdots,M$) are momentum vectors and $\phi$ is an $N$-component scalar field. The beta functions are calculated for the sine-Gordon model with multi cosine interactions. The second-order correction in the renormalization procedure is given by the two-point scattering amplitude for tachyon scattering. We show that new vertex interaction with momentum vector $k_{\ell}$ is generated from two vertex interactions with vectors $k_i$ and $k_j$ when $k_i$ and $k_j$ meet the condition $k_{\ell}=k_i\pm k_j$ called the triangle condition. Further condition $k_i\cdot k_j=\pm 1/2$ is required within the dimensional regularization method. The renormalization group equations form a set of closed equations when $\{k_j\}$ form an equilateral triangle for $N=2$ or a regular tetrahedron for $N=3$. The Wilsonian renormalization group method gives qualitatively the same result for beta functions.

The sine-Gordon model is the model of scalar field under the periodic potential. This model can be generalized in several ways. The massive chiral model is regarded as a generalization of the sine-Gordon model where the potential term Tr(g + g −1 ) is considered for g in a gauge group (Lie group) G (g ∈ G) [21][22][23]. The chiral model was generalized to include the Wess-Zumino term as the Wess-Zumino-Witten model [24][25][26][27]. The other way of generalization is to include the potential terms of high frequency modes [28]. A generalized potential term is given as where φ is a one-component scalar field and L is an integer. In the Wilson renormalization group method, the cosine potential cos((n − m)φ) is generated from cos(nφ) and cos(mφ) in the second order perturbation. Thus there will be the correction to the beta function of α n in the form α ℓ α m with n = |ℓ − m|. For the hyperbolic sine-Gordon model, α n has a correction from α ℓ and α m satisfying n = ℓ + m [29]. This kind of model can be generalized to a multicomponent scalar field. In this paper we investigate a generalized multi-component sine-Gordon model with multiple cosine potentials. The cosine vertex interaction is given by cos( ℓ k jℓ φ ℓ ) where φ = (φ 1 , · · · , φ N ) is a scalar field and k j = (k j1 , · · · , k jN ) (j = 1, · · · , M ) are momentum vectors of real numbers. k j represents the direction of oscillation of field φ. The model for M = 3 was considered in [30]. The condition to generate a new vertex interaction shown above is generalized to k n = k ℓ ± k m . This is called the triangle condition in this paper.
It was pointed out that there is a close relation between the sine-Gordon model and string propagation in a tachyon background [31]. In fact, two-vertex correction in the renormalization procedure is given by the two-point scattering amplitude for tachyon scattering in the second order perturbation theory. The multi-vertex correction will be given by the multi-point tachyon scattering amplitude.
This paper is organized as follows. In section III we present the generalized sine-Gordon model. We show the renormalization procedure based on the dimensional regularization method in section IV. We applied the Wilsonian renormalization group method to our model in section V. We consider the generalized multi-vertex sine-Gordon model and calculate the beta functions in section VI. We give summary in the last section.

II. MULTI-VERTEX SINE-GORDON MODEL
We consider a N -component real scalar field φ = (φ 1 , · · · , φ N ). The model is a d-dimensional generalized multi-vertex sine-Gordon model given by where t 0 (> 0) and α 0j (j = 1, · · · , M ) are bare coupling constants and k j (j = 1, · · · , M ) are N -component constant vectors. We use the notation (∂ µ φ) 2 = j (∂ µ φ j ) 2 and k j · φ = ℓ k jℓ φ ℓ for k j = (k j1 , · · · , k jN ). We use the Euclidean notation in this paper. The second term is the potential energy with multi cosine interactions. The dimensions of t 0 and α 0j are given as [t 0 ] = µ 2−d and [α 0j ] = µ 2 for the energy scale parameter µ. The analysis is performed near two dimensions d = 2. We introduce the renormalized coupling constants t and α j where the renormalization constants are defined as where we set that t and α j are dimensionless constants. The renormalized field φ R is introduced with the renormalization constant Z φ as follows In the following φ denotes the renormalized field φ R for simplicity. Then the Lagrangian density is given as We examine the renormalization group procedure for this model in section III and section IV. We also investigate the component dependence of renormalization in section V by generalizing the model as follows.
(6) We need some conditions so that we have one fixed point for t. For this purpose we normalize k vectors as From the two vertices with momentum vectors k i and k j , new vertex is generated with momentum k m when the triangle condition is satisfied: We assume that a set {α j } includes all vertices that will be generated from multi-vertex interactions each other. For a triangle or a regular polyhedron which is composed of equilateral triangles, M becomes finite since {k j } form a finite set. For example, we consider an equilateral triangle or a regular tetrahedron. For an equilateral triangle (N = 2, M = 3) or a regular tetrahedron (N = 3, where C(M ) is a constant depending upon M . These conditions will be explained in the following sections.

III. RENORMALIZATION BY DIMENSIONAL REGULARIZATION
We evaluate the beta functions for the multi-vertex sine-Gordon model by using the dimensional regularization method [32][33][34].

A. Tadpole renormalization of αj
The lowest order contributions to the renormalization of α j are given by tadpole diagrams. Using the expansion cos φ = 1 − 1 2 φ 2 + 1 4! φ 4 − · · · , the cosine potential is renormalized as (10) φ 2 is regularized as for d = 2 + ǫ where a mass m 0 is introduced to avoid the infrared divergence. We set Z t = 1 in the lowest order of t. We adopt that φ i φ ℓ = δ iℓ φ 2 i and φ 2 ℓ is independent of ℓ. Then the renormalization of the potential term is given as The renormalization constant Z αj is determined as Since the bare coupling constant α 0j = α j µ 2 Z αj is independent of µ, we have The beta function of α j has a zero at since k 2 j = 1.

B. Vertex-vertex interaction
We investigate the corrections to t and α j from vertexvertex interactions. The second-order contribution I (2) to the action is given by We first examine the first term denoted as I 1 : We evaluate the renormalization of cosine term by calu- is renormalized as The two-point function is written as where we introduced m 0 to avoid the infrared divergence and K 0 is the zero-th modified Bessel function.

C. Renormalization of t
The first term of I (2) 1 gives a contribution to the renormalization of the coupling constant t. Since K 0 (m 0 r) increases as r → 0, we can expand in terms of r. By us- As mentioned in section II, we consider the case where {k j } form an equilateral triangle (M = 3) or a regular tetrahedron (M = 6), and we obtain i k 2 iℓ = constant ≡ C depending on M such as C = 3/2 for M = 3 and N = 2, and C = 2 for M = 6 and N = 3. In this case Inorder to recover the kinetic term in the original action, we use the approximation Otherwise the renormalization of the kinetic term becomes complicated since we must introduce {t i } that depend on components of φ. This may not be essential for the renormalization group flow. We discuss this point later. φ 2 1 is evaluated as where a is a small cutoff. The r-integration is calculated as where c = (e γ /2) 2 . We put and since we normalize k 2 j = 1. Then we have This indicates Then we choose where we set Z αi = 1 to the lowest order of α i . Since the bare coupling constant t 0 = tµ 2−d Z t is independent of the energy scale µ, we have µ∂t 0 /∂µ = 0. This results in where we used µ∂α i /∂µ = −2(α i − t/8π), neglecting terms of the order of t 2 α 2 i , and we put a = µ −1 .

D. Vertex-vertex correction to αj
We consider the second term in I (2) 1b that contains multi-vertex interaction: Let us examine the integral given by where the cutoff a is introduced. We put t = 8π(1 + v), then we have a divergence near two dimensions when This means that two vectors k i and k j forms an equilateral triangle. When k i and k j satisfy this condition, we have Then we obtain Let k ℓ be a vector so that k i , k j and k ℓ form an equilateral triangle where Then the potential term with coefficient α ℓ has the correction as (39) We choose the renormalization constant Z α ℓ as This leads to the beta function β(α ℓ ) with correction as Since the coefficient of the correction term is dependent on cutoff parametes, we choose cm 2 0 a 2 = 1 to have There is also a contribution from the second term in I (2) where k j is replaced by −k j . In this case vertices with k i · k j = −1/2 generate a new vertex with k ℓ satisfying In the dimensional regularization method, two vertices satisfying k i · k j = ±1/2 generate a new vertex with k i ∓ k j . As a result, the beta function for α ℓ reads where the summation should take for those satisfying k ℓ = k i ± k j (i, j, ℓ = 1, · · · , N ). When k 1 , k 2 and k 3 form an equilateral triangle (M = 3), the renormalization group equations for α 1 , α 2 and α 3 are closed within three equations. When k 1 , k 2 , · · · , k 6 form a regular tetrahedron (M = 6), we again have a closed set of equations for α j (j = 1, 2, · · · , 6). In the Wilsonian method, the same beta equation for α ℓ is obtained which will be discussed in the next section. In the Wilson renormalization method, however, a new vertex k i ∓ k j is generated from any two vectors k i and k j except the case k i · k j = 0. The two-vertex correction J ij is related to the tachyon scattering amplitude in a bosonic string theory. The npoint scattering amplitude for tachyon scattering is given as [35,36] where the integration with the measure dµ is an integral over the various z i . If we assume the correspondence the z i dependence of the amplitude A 2 agrees with J ij . The vertex-vertex renormalization is given by the amplitude for tachyon scattering.

F. Renormalization group flow
For an equilateral triangle configuration of {k i } with M = 3 and N = 2, the equations read and We consider the simplified case where α i = α (i = 1, 2, 3). In this case the equations read In two dimensions d = 2, the equations become for t = 8π(1 + v). The renormalization group flow is shown in Fig. 2 for α > 0. The dotted line indicates α = −32v where µ∂α/∂µ vanishes. The asymptotic line as µ → ∞ is given by It is apparent from Fig. 2 that there is an asymmetry between positive v and negative v. This is due to the twovertex contribution. There is also an asymmetry between α > 0 and α < 0. The flow for α < 0 is obtained just by extending straight lines into the negative α region.

IV. WILSONIAN RENORMALIZATION GROUP METHOD
We investigate the renormalization of the multi-vertex sine-Gordon model by using the Wilsonian renormalization group method. We obtain the same set of equations as that in the dimensional regulatization method. The only difference is that two vertices satisfying k i · k j = 0 generate a new vertex, while k i and k j should satisfy k i · k j = ±1/2 in the dimensional regularization method.

A. Wilsonian renormalization procedure
We write the action in the following form.
where g j = α j /t and β = √ t. The field φ was scaled to βφ. We reduce the cutoff Λ in the following way: The scalar field φ = (φ 1 , · · · , φ N ) is divided into two parts as φ( The action is written as where S 1 indicates the potential term. Then the partition function is given by · conn means keeping only connected diagrams in · . Γ n (n = 1, 2, · · · ) represent contributions to the effective action.

B. Lowest order renormalization of gj
The lowest order contribution Γ 1 = − S 1 reads where the Green function G jdΛ is defined as where J 0 is the zero-th Bessel function. Up to this order, the action is renormalized to We perform the following scale transformation: where ζ is the scaling parameter for the field φ 1 . In the real space we have Then the effective action reads Here we put so that we obtain This leads to the renormalized g Rj and β R as Then we have Since β 2 = t, these results agree with those obtained by the dimensional regularization method in two dimensions.

C. Multi-vertex contributions
The second-order contribution to the effective action is We integrate out contributions with respect to φ 2 variable. For example, we use where s and s ′ take ±1. The second-order effective action Γ 2 is given as When k i · k j > 0, the second term grows large for |x − x ′ | → 0, while the first term becomes small. When k i · k j < 0, the first term instead becomes large. Hence we have where ∓ takes − when k i · k j > 0 and + for k i · k j < 0. Since the integrand is large when r is small, Γ 2 is written as where in the second term with derivative of φ 1 we keep only k i = k j term since this term otherwise becomes small due to the oscillation of cosine function. As discussed before, we use the approximation j g 2 j (∂ µ (k j · φ 1 )) 2 ≃ g 2 j C(∂ µ φ 1 ) 2 .
Then the effective action reads where A ans B are constants defined by We perform the scale transformation in eqs. (63) and (64) where the parameter ζ is chosen as Then the renormalized action is given by This results in the renormalization group equations as follows.
where the summation is taken for k i and k ℓ satisfying The resulting equations are consistent with those obtained using the dimensional regularization. Note that the sign is different because the derivative is calculated in the descending direction Λ → Λ − dΛ in the Wilsonian method. In the dimensional regularization method, the summation for g i and g ℓ is restricted to k i and k ℓ that satisfy k i · k ℓ = ±1/2. In the Wilsonian method, this condition is relaxed and new vertex is generated unles k i and k ℓ are orthogonal.

A. Renormalization of αj
As shown in the evaluation of β(t), the corrections to t are dependent on momentum parammeter {k i }. We examine this in this section. We consider the generalized Lagrangian given as (81) The potential term is renormalized to Then the correction is written as (84) This results in Then we obtain The fixed point of {t ℓ } is obtained as a zero of this equation. For an equilateral triangle where N = 2 and M = 3, we can choose {k j } as The critical value of t ℓ is obtained as In this case, the fixed point of {t ℓ } is also given by t 1c = t 2c = · · · = t 6c = 8π.

B. Renormalization of t ℓ
From the second-order perturbation, there appears the term that renormalizes the kinetic term as shown in section III. We use the following approximation here: We keep the diagonal terms (∂ µ φ ℓ ) 2 , and then I 1a in section III becomes where we put t ℓ = 8π(1 + v ℓ ) and neglect the term of the order of v ℓ . Then the kinetic term is renormalized into This leads to Then we obtain For N = 2 and M = 3, we use {k j } for an equilateral triangle, the equations for t 1 and t 2 read This is the result for the generalized multi-vertex sine-Gordon model. The qualitative property is the same as that obtained for that in section III. When α ≡ α 1 ∼ α 2 ∼ α 3 , we have with C = 3/2. This agrees with the previous result.

C. Multi-vertex contribution to αj
For the generalized model, I 1b in section III becomes The integral with respect to r becomes We consider the region near the fixed point t ℓ = 8π(1 + v ℓ ), where J ij is estimated as when k i · k j = 1/2. This indicates that (101) The potential term with two-vertex correction is obtained as where ′ i =j indicates the summation under the condition that k i ± k j = k ℓ . Then we choose Z α ℓ as The beta function up to the second order of α is given as where we set cm 2 0 a 2 = 1.

VI. SUMMARY
We investigated the multi-vertex sine-Gordon model on the basis of the renormalization group theory. We employ the dimensional regularization method and the Wilsonian renormalization group method. Two results are consistent each other. The generalized sine-Gordon model contains multiple cosine (vertex) potentials labelled by momentum parameters {k j } j=1,··· ,M . The vertex-vertex scattering amplitude is given by tachyon scattering amplitude. A new vertex k ℓ is generated from two vertex interactions k i and k j , and they are closed when momentum parameters {k j } satisfy the triangle condition that k i ± k j = k ℓ . When k i and k j are orthogonal, a new vertex is not generated. The condition k i · k j = ±1/2 is required in the dimensional regularization method.
For two-component scalar field (N = 2), {k j } should form a triangle (Wilson method) or an equilateral triangle (dimensional regularization) for M = 3. For threecomponent scalar field (N = 3), a regular tetrahedron form a closed system for M = 6. For these structures, the fixed point of {t j } is given by t 1 = t 2 = · · · = t M . A regular octahedron is also possible where there are six independent k j s and thus M = 6. For an equilateral triangle, regular tetrahedron and regular octahedron, we have j k 2 jℓ = C(M ) for j = 1, · · · , M where we impose the normalization ℓ k 2 jℓ = 1. We expect that there exist crystal structures in higher dimensions N ≥ 3 satisfying j k 2 jℓ = const. for any j. The beta function of α ℓ is generalized to include the product α i α j for which k i ± k j = k ℓ is satisfied. This term is a non-trivial contribution compared to the conventional sine-Gordon model. The beta function of t ℓ has also contributions proportional to α 2 j . These terms are positive and thus do not change the renormalization group flow of t ℓ . The additional terms to β(α ℓ ) change the flow of (α ℓ , t j ) qualitatively. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Grant No. 17K05559).