Dimensional Regularization Approach to the Renormalization Group Theory of the Generalized Sine-Gordon Model

We present the dimensional regularization approach to the renormalization group theory of the generalized sine-Gordon model. The generalized sine-Gordon model means the sine-Gordon model with high frequency cosine modes. We derive renormalization group equations for the generalized sine-Gordon model by regularizing the divergence based on the dimensional method. We discuss the scaling property of renormalization group equations. The generalized model would present a new class of scaling property.


Introduction
The sine-Gordon model is an interesting model and plays an important role in physics [1][2][3][4][5][6][7][8][9][10][11][12][13].There are many phenomena that are related to the sine-Gordon model.In this sense, the sine-Gordon model has universality.In the weak coupling phase the sine-Gordon model is perturbatively equivalent to the massive Thirring model [1,[14][15][16].The two-dimensional (2D) sine-Gordon model describes a crossover between weak coupling region and strong coupling region.The renormalization equations are the same as those for the Kosterlitz-Thouless transition of the 2D classical XY model [17][18][19].The 2D sine-Gordon model is mapped to the Coulomb gas model where particles interact with each other through logarithmic interaction [4,20,21].The Kondo problem belongs to the same universality class where the renormalization group equations are given by the same equations for the 2D sine-Gordon model [20][21][22][23][24][25][26][27].The renormalization group equations in the Kondo problem was derived before those by Kosterlitz and Thouless.The one-dimensional Hubbard model is mapped to the 2D sine-Gordon model by using a bosonization method [28][29][30][31], where the Hubbard model is an important model that describes the metal-insulator transition and high-temperature superconductivity [32][33][34][35][36][37][38][39].The sine-Gordon model appears in a multiband superconductor where the Nambu-Goldstone modes become massive due to the Josephson couplings [40][41][42][43][44][45][46][47].The Josephson plasma oscillation in layered high-temperature superconductors was analyzed based on the sine-Gordon model [48].In a series of papers [41-43, 45, 46] we introduced the sine-Gordon model into the study of superconductivity and examined significant excitation modes in superconductors.A generalization from U(1) to a compact continuous group G for the sine-Gordon model was also investigated [49] where the sine-Gordon model considered in this paper and in references cited above is a model with U(1) group.
In this paper, we investigate the renormalization group theory for the 2D generalized sine-Gordon model by using the dimensional regularization method to regularize the divergence [50][51][52].Here the generalized sine-Gordon model is a sine-Gordon model that includes high frequency cosine potential terms such as cos() for an integer n.The renormalization of the generalized sine-Gordon model was investigated [53] by the Wegner-Houghton method [54] and by the functional renormalization group method [55].We use the dimensional regularization method in deriving the renormalization group equation for the generalized sine-Gordon model.The divergence is regularized near two dimensions by putting the dimension  = 2 + .The divergent part of integral is evaluated as a pole in the form 1/.This is called 2 Advances in Mathematical Physics the minimal subtraction method.Then the beta function for the coupling constant is derived.

Lagrangian
Let us consider a real scalar field .The Lagrangian of the generalized sine-Gordon model is given by where where   and   are renormalization constants. and   are dimensionless constants by virtue of the energy scale .We define the renormalized field   by where   is the renormalization constant for the field .The Lagrangian with renormalized quantities is written as where  denotes the renormalized field   .The second term represents the interaction of the field  as seen by expanding cos(√  ) as a power series.There is the other representation of interaction parameters.We can absorb the parameter  in the definition of the field  and the parameter   .In this case, field  in the interaction term includes the parameter in the form cos(√  ) where  = √.We will obtain the same result since it does not depend on the representation.

Renormalization of 𝛼 𝑛
We consider the renormalization of   up to the lowest order of   .By considering tadpole diagrams in Figure 1, the cosine function is renormalized to Since the expectation value ⟨ 2 ⟩ diverges, we regularize it using the dimensional regularization method: for  = 2 +  where  0 is introduced to avoid infrared divergence, Ω  is the solid angle in  dimensions and   was put as 1.In order to remove the divergence, the constant   is determined as follows: Since the bare coupling constant  0 is independent of , we have   0 / = 0.This results in We set   = 1 up to the lowest order of   , so that we have  / = ( − 2).The beta function for   at the lowest order in   is given by (  ) has a zero at  =   : for  = 2.There is a fixed point of  for each .

Renormalization of 𝑡
There is an effect of renormalization on the coupling constant  that is the correction to the kinetic term.Let us consider the two-point function Γ().The bare lowest order two-point function is given by This corresponds to the kinetic part of the bare Lagrangian: 4.1.Real Space Formulation.The lowest order correction to the two-point function is given by a second-order term for  ℓ (ℓ = 1, 2, ⋅ ⋅ ⋅ ) such as     cos(√  ()) cos(√  (  )).
From the formula cos  1 cos  2 = (1/2)(cos( 1 +  2 ) + cos( 1 −  2 )), the correction to the action comes from where we consider connected contributions.By taking into account the contribution of tadpole diagrams, this reduces to The expectation value ⟨()()⟩ is given by where  0 is the 0th modified Bessel function.Because  0 () increases divergently as  approaches zero,   is approximated as where we put   =  + .The cosine function cos(√  ( − )()) would oscillate as a function of , the contribution for  ̸ =  will be small.Thus, we consider only the contributions with  = : We extract the divergent term in   .There may be two ways to do this.We discuss these methods in the following.
(1) In the first method, we regularize ⟨ 2 ⟩ by introducing a cutoff  in the real space: by replacing  0 ( 0 ) with  0 ( 0 √  2 +  2 ).By using the asymptotic relation  0 () ≈ − − ln(/2) with the Euler constant , the integral with respect to  is performed as follows [49]: near  = 2 where we set  = (  /2) 2 .We consider the case where  is close to the critical value   = 8/ 2 : where V  represents the deviation from the critical point.In the lowest order of V  , we have Then we obtain The constant   was absorbed for the renormalization of   .Then, by taking the sum from each term, the kinetic part L (0)  is renormalized to This indicates that we choose and   appear as a ratio   /  in this order, and then the coupling constant  is renormalized as  0 =  2−   /  or we can choose   = 1.The equation   0 / = 0 results in Lastly, we put  =  −1 to obtain The numerical coefficient is not important and this depends on the choice of the cutoff .
(2) In the second way, the divergence comes from ⟨ 2 ⟩ where we adopt that the integral with respect to  is finite.This treatment is similar to that in [31] where the Wilson The contributions to the two-point function Γ (2) () up to the order of     .renormalization group method was used.The correction   is written as In order to let the integral for  be dimensionless, we change the variable  =  and put by introducing a cutoff in the integral.Then, we have This results in the same beta function () with the numerical factor being slightly different: 4.2.Momentum Space Formulation.In the momentum space, we evaluate the two-point function by calculating the diagrams in Figure 2 [6].This set of diagrams gives the selfenergy Σ().Σ() is written as a sum of Σ  () that comes from the interaction term cos() cos().The diagrams in Figure 2 are summed up to give Advances in Mathematical Physics 5 where we put  0 () =   ⟨ ()  (0)⟩ .
When  ≈ 2, we put  = 2(1+V 2 ) and  2 = 2V 2 to obtain In this region,  1 acts as a perturbation to the scaling equation of the conventional sine-Gordon model.We show the renormalization group flow as  increases in Figure 3.

Figure 1 :
Figure 1: One-loop contributions to the renormalization of   .
is a bare real scalar field and  0 and  0 are bare coupling constants.The second term indicates the potential energy of the scalar field .The generalized sine-Gordon model contains high frequency terms such as cos() (n = 1, 2, ⋅ ⋅ ⋅ ).We write the renormalized coupling constants as  and   , respectively.We adopt that  > 0 and   ≥ 0.   for some  may be zero, but at least one   should be positive (nonzero).The dimensions of  and   are given as [] =  2− and [  ] =  2 where  is a parameter representing the energy scale.The scalar field  is dimensionless.The relations between bare and renormalized quantities are given by 1 ,  1 =  1 /4and  2 =  2 /2.The equations read