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

Authors: Takashi Yanagisawa
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

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.


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## Body

## 1. Introduction

The sine-Gordon model is an interesting model and plays an important role in physics [1–13]. 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, 14–16]. 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 [17–19]. 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 [20–27]. 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 [28–31], where the Hubbard model is an important model that describes the metal-insulator transition and high-temperature superconductivity [32–39]. The sine-Gordon model appears in a multiband superconductor where the Nambu-Goldstone modes become massive due to the Josephson couplings [40–47]. 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 [50–52]. Here the generalized sine-Gordon model is a sine-Gordon model that includes high frequency cosine potential terms such as cos⁡nϕ 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 d=2+ϵ. The divergent part of integral is evaluated as a pole in the form 1/ϵ. This is called the minimal subtraction method. Then the beta function for the coupling constant is derived.


## 2. Lagrangian

Let us consider a real scalar fieldϕ. The Lagrangian of the generalized sine-Gordon model is given by(1)L=12t0∂μϕB2+1t0∑nαn0cos⁡nϕB,where ϕB is a bare real scalar field and t0 and α0n are bare coupling constants. The second term indicates the potential energy of the scalar field ϕ. The generalized sine-Gordon model contains high frequency terms such as cos⁡nϕ (n = 1, 2, ⋯). We write the renormalized coupling constants as t and αn, respectively. We adopt that t>0 and αn≥0. αm for some m may be zero, but at least one αn should be positive (nonzero). The dimensions of t and αn are given as t=μ2-d and αn=μ2 where μ is a parameter representing the energy scale. The scalar field ϕ is dimensionless. The relations between bare and renormalized quantities are given by(2)t0=tμ2-dZt,αn0=αnμ2Zαn,where Zt and Zαn are renormalization constants. t and αn are dimensionless constants by virtue of the energy scale μ. We define the renormalized field ϕR by(3)ϕB=ZϕϕR,where Zϕ is the renormalization constant for the field ϕ. The Lagrangian with renormalized quantities is written as(4)L=μd-2Zϕ2tZtϑμϕ2+∑nμdαnZαntZtcos⁡nZϕϕ,where ϕ denotes the renormalized field ϕR. The second term represents the interaction of the field ϕ as seen by expanding cos⁡nZϕϕ as a power series. There is the other representation of interaction parameters. We can absorb the parameter t in the definition of the field ϕ and the parameter αn. In this case, field ϕ in the interaction term includes the parameter in the form cos⁡nZϕβϕ where β=t. We will obtain the same result since it does not depend on the representation.


## 3. Renormalization ofαn

We consider the renormalization ofαn up to the lowest order of αn. By considering tadpole diagrams in Figure 1, the cosine function is renormalized to (5)cos⁡nZϕϕ⟶1-12n2Zϕϕ2+⋯cos⁡nZϕϕ=exp⁡-12n2Zϕϕ2cos⁡nZϕϕ.Since the expectation value ϕ2 diverges, we regularize it using the dimensional regularization method:(6)Zϕϕ2=tμ2-dZt∫ddk2πd1k2+m02=-tϵΩd2πd,for d=2+ϵ where m0 is introduced to avoid infrared divergence, Ωd is the solid angle in d dimensions and Zt was put as 1. In order to remove the divergence, the constant Zαn is determined as follows:(7)Zαn=1-n22t1ϵΩd2πd.Since the bare coupling constant αn0 is independent of μ, we have μ∂αn0/∂μ=0. This results in(8)μ∂αn∂μ=-2αn-αnμ∂ln⁡Zαn∂μ.We set Zt=1 up to the lowest order of αn, so that we have μ∂t/∂μ=d-2t. The beta function for αn at the lowest order in αn is given by(9)βαn≡μ∂αn∂μ=-2αn+n2tαn12Ωd2πd.βαn has a zero at t=tcn:(10)tcn=8πn2,for d=2. There is a fixed point of t for each n.Figure 1
One-loop contributions to the renormalization ofαn.


## 4. Renormalization oft

There is an effect of renormalization on the coupling constantt that is the correction to the kinetic term. Let us consider the two-point function Γp. The bare lowest order two-point function is given by(11)ΓB0p=1t0p2=1tμ2-dZtp2,This corresponds to the kinetic part of the bare Lagrangian:(12)LB0=μd-2Zϕ2tZt∂μϕ2.


### 4.1. Real Space Formulation

The lowest order correction to the two-point function is given by a second-order term forαl (l=1,2,⋯) such as αnαmcos⁡nZϕϕxcos⁡mZϕϕx′. From the formula cos⁡θ1cos⁡θ2=1/2cos⁡θ1+θ2+cos⁡θ1-θ2, the correction to the action comes from(13)Inm≡-12!12μdtZt2ZαnZαmαnαm∫ddx∫ddx′cos⁡nZϕϕx-mZϕϕx′,where we consider connected contributions. By taking into account the contribution of tadpole diagrams, this reduces to(14)Inm=-14ZαnZαmαnαm∫ddx∫ddx′1-12Zϕn2ϕx2+m2ϕx′2-2nmϕxϕx′+⋯×μdtZt2cos⁡nZϕϕx-mZϕϕx′≅-14ZαnZαmαnαm∫ddx∫ddx′exp⁡-12Zϕn2ϕx2+m2ϕx′2++nmZϕϕxϕx′μdtZt2cos⁡nZϕϕx-mZϕϕx′.The expectation value ϕxϕy is given by(15)ϕxϕy=tμ2-dZtZϕ∫ddp2πdeip·x-yp2+m02=tμ2-dZtZϕΩd2πdK0m0x-y,where K0 is the 0th modified Bessel function. Because K0x increases divergently as x approaches zero, Inm is approximated as(16)Inm≅-14ZαnZαmαnαm∫ddx∫ddrexp⁡-12Zϕn2+m2ϕ2+nmtΩd2πdK0m0rμdtZt2cos⁡Zϕn-mϕx1-12m2Zϕ∇xϕ·r2,where we put x′=x+r. The cosine function cos⁡Zϕn-mϕx would oscillate as a function of x, the contribution for n≠m will be small. Thus, we consider only the contributions with n=m:(17)Inm≅14αn2Zαn2μdtZt2∫ddxddr12n2Zϕ∇xϕ212r2exp⁡-Zϕn2ϕ2+n2tΩd2πdK0m0r.We extract the divergent term in Inn. There may be two ways to do this. We discuss these methods in the following.( 1 ) In the first method, we regularize ϕ2 by introducing a cutoff a in the real space:(18)Zϕϕ2=tμ2-dZtΩd2πdK0m0a,by replacing K0m0r with K0m0r2+a2. By using the asymptotic relation K0z≈-γ-ln⁡z/2 with the Euler constant γ, the integral with respect to r is performed as follows [49]:(19)Jn≡∫ddrr2exp⁡n2tΩd2πdK0m0r2+a2≈Ωd∫drrd+11cm02r2+a2n2t/4π,near d=2 where we set c=eγ/22. We consider the case where t is close to the critical value tcn=8π/n2:(20)n2t8π=1+vn,where vn represents the deviation from the critical point. In the lowest order of vn, we have(21)Jn=Ωdcm02-212-d+Ovn=-Ωdcm02-21ϵ+Ovn.Then we obtain(22)Inn≈-18αn2Zαn2μdtZt2exp⁡n2tZt14πlog⁡cm02a2Ωdcm02-21ϵ∫ddx12n2Zϕ∂μϕ2+Ovn≈-n232αn2μd+2a41ϵ∫ddxμd-2Zϕ2tZtn2∂μϕ2+Ovn.The constant Zαn was absorbed for the renormalization of αn. Then, by taking the sum from each term, the kinetic part LB(0) is renormalized to(23)LB2=μd-2Zϕ2tZt1-∑nn432αn2μd+2a41ϵ∂μϕ2.This indicates that we choose(24)ZϕZt=1+∑nn432αn2μd+2a41ϵ.Zϕ and Zt appear as a ratio Zϕ/Zt in this order, and then the coupling constant t is renormalized as t0=tμ2-dZt/Zϕ or we can choose Zϕ=1. The equation μ∂t0/∂μ=0 results in(25)βt≡μ∂t∂μ=d-2t+∑nn432a4tϵ2αnμ∂αn∂μμd+2+d+2αn2μd+2=d-2t+∑nn432tαn2μd+2a4+Ot2.Lastly, we put a=μ-1 to obtain(26)βt=d-2t+∑nn432tαn2+Ot2.The numerical coefficient is not important and this depends on the choice of the cutoff a.( 2 ) In the second way, the divergence comes from ϕ2 where we adopt that the integral with respect to r is finite. This treatment is similar to that in [31] where the Wilson renormalization group method was used. The correction Inn is written as(27)Inn=18αn2Zαn2μdtZt2exp⁡-n2Zϕϕ2∫ddrr2exp⁡n2tΩd2πdK0m0r∫ddx12n2Zϕ∂μϕ2≈-n48αn2μd+212πϵ∫ddrr2exp⁡n2t2πK0m0r∫ddxμd-2Zϕ2tZt∂μϕ2.In order to let the integral for r be dimensionless, we change the variable r=au and put(28)An=∫dduu2exp⁡n2t2πK0m0au2+1,by introducing a cutoff in the integral. Then, we have (29)Inn≅-An16πn4αn2μd+2ad+21ϵ∫ddxμd-2Zϕ2tZt∂μϕ2.This results in the same beta function βt with the numerical factor being slightly different:(30)βt=d-2t+∑nAn16πn4tαn2.


### 4.2. Momentum Space Formulation

In the momentum space, we evaluate the two-point function by calculating the diagrams in Figure2 [6]. This set of diagrams gives the self-energy Σp. Σp is written as a sum of Σnmp that comes from the interaction term cos⁡nϕcos⁡mϕ. The diagrams in Figure 2 are summed up to give(31)Σnmp=nmαnαmZαnZαmμdtZt2∫ddxeip·xsinh⁡nmG0x-nmG0x-cosh⁡nmG0x-1,where we put(32)G0x=Zϕϕxϕ0.Since G0x is divergently large as x→0, Σnm is approximated as(33)Σnmp=12nmαnαmZαnZαmμdtZt2∫ddxeip·x-1exp⁡nmG0x.Using the expansion eip·x=1+ip·x-1/2p·x2+⋯, we keep the p2 term. By using the formula K0x≈-γ-log⁡x/2 for small x, Σnmp is written as(34)Σnmp=-14nmαnαmZαnZαmμdtZt2∫ddxxd+11cm02x2+a2nmt/4π=-18p2nmαnαmZαnZαmμdtZt2Ωd∫dxxd+11cm02x2+a2nmt/4π.The integral diverges when t=tcn=8π/n2 and n=m. Then we consider the case n=m, which gives the correction to the two-point function ΓB2n=-Σnn when t≈tcn as follows:(35)ΓB2np=-18p2n2αn2Zαn2μdtZt2Ωdcm02-21ϵ≅-1tμ2-dZtp2132n4αn2μd+2cm02-21ϵ+Ovn,where we set t≈8π/n2·1+vn. This term mainly comes from the region where t≈tcn. The two-point function up to this order is(36)ΓB2p=1tμ2-dZtp2-p2∑nn432αn2μd+2cm02-21ϵ.The renormalized two-point function is given as ΓR(2)p=ZϕΓB(2)p. The renormalization constants are determined as shown above and thus we obtain the same renormalization group equation.Figure 2
The contributions to the two-point functionΓ(2)p up to the order of αnαm.


## 4.1. Real Space Formulation

The lowest order correction to the two-point function is given by a second-order term forαl (l=1,2,⋯) such as αnαmcos⁡nZϕϕxcos⁡mZϕϕx′. From the formula cos⁡θ1cos⁡θ2=1/2cos⁡θ1+θ2+cos⁡θ1-θ2, the correction to the action comes from(13)Inm≡-12!12μdtZt2ZαnZαmαnαm∫ddx∫ddx′cos⁡nZϕϕx-mZϕϕx′,where we consider connected contributions. By taking into account the contribution of tadpole diagrams, this reduces to(14)Inm=-14ZαnZαmαnαm∫ddx∫ddx′1-12Zϕn2ϕx2+m2ϕx′2-2nmϕxϕx′+⋯×μdtZt2cos⁡nZϕϕx-mZϕϕx′≅-14ZαnZαmαnαm∫ddx∫ddx′exp⁡-12Zϕn2ϕx2+m2ϕx′2++nmZϕϕxϕx′μdtZt2cos⁡nZϕϕx-mZϕϕx′.The expectation value ϕxϕy is given by(15)ϕxϕy=tμ2-dZtZϕ∫ddp2πdeip·x-yp2+m02=tμ2-dZtZϕΩd2πdK0m0x-y,where K0 is the 0th modified Bessel function. Because K0x increases divergently as x approaches zero, Inm is approximated as(16)Inm≅-14ZαnZαmαnαm∫ddx∫ddrexp⁡-12Zϕn2+m2ϕ2+nmtΩd2πdK0m0rμdtZt2cos⁡Zϕn-mϕx1-12m2Zϕ∇xϕ·r2,where we put x′=x+r. The cosine function cos⁡Zϕn-mϕx would oscillate as a function of x, the contribution for n≠m will be small. Thus, we consider only the contributions with n=m:(17)Inm≅14αn2Zαn2μdtZt2∫ddxddr12n2Zϕ∇xϕ212r2exp⁡-Zϕn2ϕ2+n2tΩd2πdK0m0r.We extract the divergent term in Inn. There may be two ways to do this. We discuss these methods in the following.( 1 ) In the first method, we regularize ϕ2 by introducing a cutoff a in the real space:(18)Zϕϕ2=tμ2-dZtΩd2πdK0m0a,by replacing K0m0r with K0m0r2+a2. By using the asymptotic relation K0z≈-γ-ln⁡z/2 with the Euler constant γ, the integral with respect to r is performed as follows [49]:(19)Jn≡∫ddrr2exp⁡n2tΩd2πdK0m0r2+a2≈Ωd∫drrd+11cm02r2+a2n2t/4π,near d=2 where we set c=eγ/22. We consider the case where t is close to the critical value tcn=8π/n2:(20)n2t8π=1+vn,where vn represents the deviation from the critical point. In the lowest order of vn, we have(21)Jn=Ωdcm02-212-d+Ovn=-Ωdcm02-21ϵ+Ovn.Then we obtain(22)Inn≈-18αn2Zαn2μdtZt2exp⁡n2tZt14πlog⁡cm02a2Ωdcm02-21ϵ∫ddx12n2Zϕ∂μϕ2+Ovn≈-n232αn2μd+2a41ϵ∫ddxμd-2Zϕ2tZtn2∂μϕ2+Ovn.The constant Zαn was absorbed for the renormalization of αn. Then, by taking the sum from each term, the kinetic part LB(0) is renormalized to(23)LB2=μd-2Zϕ2tZt1-∑nn432αn2μd+2a41ϵ∂μϕ2.This indicates that we choose(24)ZϕZt=1+∑nn432αn2μd+2a41ϵ.Zϕ and Zt appear as a ratio Zϕ/Zt in this order, and then the coupling constant t is renormalized as t0=tμ2-dZt/Zϕ or we can choose Zϕ=1. The equation μ∂t0/∂μ=0 results in(25)βt≡μ∂t∂μ=d-2t+∑nn432a4tϵ2αnμ∂αn∂μμd+2+d+2αn2μd+2=d-2t+∑nn432tαn2μd+2a4+Ot2.Lastly, we put a=μ-1 to obtain(26)βt=d-2t+∑nn432tαn2+Ot2.The numerical coefficient is not important and this depends on the choice of the cutoff a.( 2 ) In the second way, the divergence comes from ϕ2 where we adopt that the integral with respect to r is finite. This treatment is similar to that in [31] where the Wilson renormalization group method was used. The correction Inn is written as(27)Inn=18αn2Zαn2μdtZt2exp⁡-n2Zϕϕ2∫ddrr2exp⁡n2tΩd2πdK0m0r∫ddx12n2Zϕ∂μϕ2≈-n48αn2μd+212πϵ∫ddrr2exp⁡n2t2πK0m0r∫ddxμd-2Zϕ2tZt∂μϕ2.In order to let the integral for r be dimensionless, we change the variable r=au and put(28)An=∫dduu2exp⁡n2t2πK0m0au2+1,by introducing a cutoff in the integral. Then, we have (29)Inn≅-An16πn4αn2μd+2ad+21ϵ∫ddxμd-2Zϕ2tZt∂μϕ2.This results in the same beta function βt with the numerical factor being slightly different:(30)βt=d-2t+∑nAn16πn4tαn2.


## 4.2. Momentum Space Formulation

In the momentum space, we evaluate the two-point function by calculating the diagrams in Figure2 [6]. This set of diagrams gives the self-energy Σp. Σp is written as a sum of Σnmp that comes from the interaction term cos⁡nϕcos⁡mϕ. The diagrams in Figure 2 are summed up to give(31)Σnmp=nmαnαmZαnZαmμdtZt2∫ddxeip·xsinh⁡nmG0x-nmG0x-cosh⁡nmG0x-1,where we put(32)G0x=Zϕϕxϕ0.Since G0x is divergently large as x→0, Σnm is approximated as(33)Σnmp=12nmαnαmZαnZαmμdtZt2∫ddxeip·x-1exp⁡nmG0x.Using the expansion eip·x=1+ip·x-1/2p·x2+⋯, we keep the p2 term. By using the formula K0x≈-γ-log⁡x/2 for small x, Σnmp is written as(34)Σnmp=-14nmαnαmZαnZαmμdtZt2∫ddxxd+11cm02x2+a2nmt/4π=-18p2nmαnαmZαnZαmμdtZt2Ωd∫dxxd+11cm02x2+a2nmt/4π.The integral diverges when t=tcn=8π/n2 and n=m. Then we consider the case n=m, which gives the correction to the two-point function ΓB2n=-Σnn when t≈tcn as follows:(35)ΓB2np=-18p2n2αn2Zαn2μdtZt2Ωdcm02-21ϵ≅-1tμ2-dZtp2132n4αn2μd+2cm02-21ϵ+Ovn,where we set t≈8π/n2·1+vn. This term mainly comes from the region where t≈tcn. The two-point function up to this order is(36)ΓB2p=1tμ2-dZtp2-p2∑nn432αn2μd+2cm02-21ϵ.The renormalized two-point function is given as ΓR(2)p=ZϕΓB(2)p. The renormalization constants are determined as shown above and thus we obtain the same renormalization group equation.Figure 2
The contributions to the two-point functionΓ(2)p up to the order of αnαm.


## 5. Renormalization Group Flow

Let us consider the case with two parametersα1 and α2. The renormalization group equations are(37)μ∂α1∂μ=-2α11-18πt,(38)μ∂α2∂μ=-2α21-12πt,(39)μ∂t∂μ=d-2t+132tα12+18tα22.We have the critical value t=tc1=8π for α1 and t=tc2=2π for α2. The parameter t is an increasing function of μ in two dimensions d=2. The space t,α1,α2 may be divided into four regions which are classified by the values to which the pair α1,α2 is renormalized as μ increases. We call them regions I, II, III, and IV:(40)I:α1,α2→∞,∞,(41)II:α1,α2→0,∞,(42)III:α1,α2→∞,0,(43)IV:α1,α2→0,0.In region t≈8π, we put t=8π1+v1, x1=2v1, y1=α1/4and y2=α2/2. The equations read(44)μ∂x1∂μ=y12+y22,(45)μ∂y1∂μ=x1y1,(46)μ∂y2∂μ=6y2.When y2 is small, the equations reduce to those of the conventional sine-Gordon model (Kosterlitz-Thouless transition).Whent≈2π, we put t=2π1+v2 and x2=2v2 to obtain(47)μ∂x2∂μ=y12+y22,(48)μ∂y1∂μ=-32y1,(49)μ∂y2∂μ=x2y2.In this region, y1 acts as a perturbation to the scaling equation of the conventional sine-Gordon model. We show the renormalization group flow as μ increases in Figure 3.Figure 3
Renormalization group flow asμ→∞ for α1 and α2.


## 6. Summary

We have discussed the dimensional regularization approach to the renormalization group theory of the generalized sine-Gordon model. There are multiple critical points for the coupling constantt given as tcn=8π/n2. In the case where t is close to tcn for some n, the renormalization group equations are approximated by those for the sine-Gordon model with single-cosine potential (conventional sine-Gordon model). A nontrivial simple generalized model is the sine-Gordon model with α1 and α2. When t is tc1=8π, α2 acts as a perturbation for t and α1. The renormalization flow as μ→∞ or μ→0 depends on an initial set of parameters t, α1, and α2. This can be viewed as a competition between two interactions α1 and α2. This may lead to a generalization of the Kosterlitz-Thouless transition, the crossover phenomenon in the Kondo effect, and other phenomena. In the Kondo problem, the appearance of logarithmic singularity [22, 56, 57] suggested the renormalizability of the model. In materials with many magnetic impurities, the interaction between magnetic impurities, called the RKKY interaction [58–60], should be considered. In this case the renormalization group flow is drawn on a two-dimensional plane of two parameters. There may be a relation to the generalized sine-Gordon model.



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