Exact renormalization group and Sine Gordon theory

The exact renormalization group is used to study the RG flow of quantities in field theories. The basic idea is to write an evolution operator for the flow and evaluate it in perturbation theory. This is easier than directly solving the differential equation. This is illustrated by reproducing known results in four dimensional ϕ4 field theory and the two dimensional Sine-Gordon theory. It is shown that the calculation of beta function is somewhat simplified. The technique is also used to calculate the c-function in two dimensional Sine-Gordon theory. This agrees with other prescriptions for calculating c-functions in the literature. If one extrapolates the connection between central charge of a CFT and entanglement entropy in two dimensions, to the c-function of the perturbed CFT, then one gets a value for the entanglement entropy in Sine-Gordon theory that is in exact agreement with earlier calculations (including one using holography) in arXiv:1610.04233.


Introduction
The exact renormalisation group (ERG), first written down by Wilson [1][2][3] has been an object of much study. It has been developed further [4] and different versions suitable for different purposes have been written down since then [5][6][7][8]. There are a large number of good reviews [9][10][11][12][13]. A lot of work has been done on the RG of the Sine-Gordon model over the last few years and many computations have been carried out analytically and numerically .

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This paper primarily studies the application of ERG mainly to two dimensional field theories -the emphasis being on a simple way of writing down the solution to the ERG in terms of an evolution operator. We will reproduce some known results and also obtain a new result on the flow of the c-function in Sine-Gordon theory. While the main application of the ERG has been in the study of critical phenomena -to obtain the numerical value of critical exponents, our motivation comes from string theory. In the context of string theory the renormalisation group has been used as a formal tool. Recently, the ERG was used to obtain the equations of motion for the fields of the string somewhat as in string field theory [17]. In this approach string propagation in a general background is described as a completely general two dimensional field theory -all relevant, irrelevant and marginal terms are included. This is a natural generalization of the idea that a conformal field theory describes a consistent string background. 1 The condition of conformal invariance is imposed on the action. Thus the exact renormalisation group equation for this two dimensional theory is written down and the fixed point equations for the couplings are identified with the space time equations of motion of the background fields. One has to further generalise the original RG approach to obtain equations that are gauge invariant. The new ingredient is the use of loop variables. In order to make the equations gauge invariant the two dimensional field theory is written in terms of loop variables [18]. Loop variables have also been incorporated into the ERG -and gauge invariant and interacting equations have been written down. Furthermore these equations are background independent [17].
Again within string theory, but now in the context of AdS/CFT correspondence the idea of the renormalisation group has emerged in the guise of holographic RG [19,20]. The RG flow of quantities has been equated with the evolution of the holographic dual bulk fields in the radial direction. Thus a flow of renormalised coupling constants in the boundary is compared to the flow of the bulk field, which also requires renormalization. Many details of this comparison have been worked out in [21]. In field theory there are quantities such as the c-function of Zamolodchikov in two dimensions [23] and the c and a-functions in four dimensions [33] that are monotonic along the flow. There have been attempts to find analogous quantities in the holographic dual in the bulk. One such quantity is the entanglement entropy which has also been shown to be monotonic along the flowboth in the field theory and its holographic dual. (Although the precise connection with Zamolodchikov's c-function is not established.) Besides being interesting due to the connection with string theory, two dimensional models have some advantages as an arena where these ideas can be developed. They are simpler to work with and their holographic dual AdS 3 equations are often exactly solvable. This motivates us to explore two dimensional field theories using ERG. An interesting and very non trivial field theory is the Sine-Gordon theory. The Sine-Gordon β-functions in fact are closely related to equations of motion of the bosonic string tachyon [16]. This is a special case of the connection to string theory mentioned above. In the context of critical phenomena also this model has been related to a very interesting two dimensional model -the X-Y model. The X-Y model has an interesting phase transition first noticed by

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Kosterlitz and Thouless. It is possible to rewrite the X-Y model as a Sine-Gordon model. This theory has been studied in great detail in [22] who obtained the phase diagram as well as the Kosterlitz-Thouless RG flow equations using continuum field theory techniques. They also showed that the model is renormalizable when physical quantities are written as a power series in terms of two coupling constants. 2 In this paper we use the ERG to obtain the β function equations of Sine-Gordon theory using the ERG. A particular form of the ERG due to Polchinski is used here. This ERG is reformulated as a linear evolution operator. Although this reformulation has been noticed [5,6], it has not received much attention in practice. We show that it is very convenient to work out the flow of objects in a systematic perturbation series. In the usual continuum calculations β-functions are calculated as a byproduct of the renormalization program. As first explained by Wilson [2], when one obtains the flow of a marginal coupling, in the limit that the UV cutoff is taken to infinity, the β-function has the property that it depends only on the value of the coupling and not explicitly on the scale. This also implies that the logarithmic divergence has the information about the β-function and higher orders in the logarithm are determined by the coefficient of the leading logarithmic divergence. Thus if we have an evolution equation one needs to only evaluate the leading divergence. Furthermore this is different from actually solving the ERG equation which gives a coupled differential equation involving an infinite number of couplings. The process of eliminating the irrelevant couplings and solving for the marginal coupling is automatically implemented during the perturbative evaluation of the evolution operator.
Thus mathematically one can imagine a set of coupled recursion equations [2] for a marginal coupling g l , a relevant coupling µ l and an irrelevant coupling w l obtained in a blocking transformation that implements the RG. We reproduce a summary of the discussion in [2](The factors of 4 and 1/4 are illustrative): Here N g , N w , N µ are the nonlinear terms. As explained in [2] one can reorganize the equations and solve them iteratively so that it depends on w 0 (initial condition for w l ) and µ L (final value of µ l ) and then one finds that on solving this iteratively, and when 1 ≪ l ≪ L is very large, so the memory of the initial conditions have been lost, one can set µ L = w 0 = 0 and obtain a recursion equation for g l alone The crucial point is that in this limit V (g l ) has no explicit dependence on l. One can now extract from this a β-function β g = dg dt (l is replaced by a continuous variable t) which depends only on g(t).

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Now imagine using the evolution equation to obtain g(t + τ ) starting from g(t). One obtains a series of the form Now dg(t) dt = β(g(t)). Thus Thus when t = 0, τ = ln Λ 0 Λ is what we call the logarithmic divergence in perturbation theory. What we are seeing is that the leading term decides the β-function and the higher powers of τ are fixed in terms of the leading term. 3 The application of the evolution operator in powers of the evolution Hamiltonian, gives us a series as above in τ . It automatically gives the solution of the ERG recursion equations and one can extract a power series for the evolution of the marginal coupling. Thus β -functions are obtained in a simple way without worrying about the technicalities of renormalization.
We illustrate this method with some examples such as the central charge of a free scalar field theory and the flow of coupling in φ 4 theory in four dimensions. We then apply it to the more interesting case of the Sine-Gordon theory. We find that the equations obtained are consistent with those obtained in [22]. While the precise coefficients are not the same the combination of coefficients identified in [22] as being universal matches exactly. In addition to flow of couplings, one can study the flow of the c-function [23][24][25][26]. In particular we do the calculation of the c-function for the Sine-Gordon theory. Recently the entanglement entropy of this theory has been calculated both in the field theory and in the holographic dual and the answers are shown to agree to lowest order [27]. The central charge calculation done here also gives results in exact agreement with these calculationsif we assume that the relation between entanglement entropy and central charge function persists at least to lowest non trivial order away from the fixed point. This is a first step towards an understanding of a precise connection between the ERG in the boundary theory and Holographic RG in the bulk. This paper is organised as follows: section 2 gives some background material on the ERG. In section 3 we describe the evolution operator approach and illustrate it with a calculation of the c-function in a free massive scalar field theory. In section 4 we illustrate the method by calculating the β-function for φ 4 field theory in four dimensions. In section 5 the β-functions for sine-Gordon theory are calculated. In section 6 the c-function for the sine Gordon theory is calculated. Section 7 contains a summary and some conclusions.

Polchinski equation
The original Wilsonian form of the ERG had roughly the following structure (Ġ = ∂G ∂τ ): If we substitute ψ = e − 1 2 G −1 y 2 ψ ′ we get an equation: Again in terms of S it becomes an equation in the form first written by Polchinski [20] ∂S ∂τ = − 1 2Ġ In these equations one can replace y by φ(p) and easily generalise to field theory. In a field theory RG τ is the logarithm of the ratio of scales: the short distance cutoff a is changed to ae τ . In a field theory action 1 2 G −1 y 2 would stand for the kinetic term (and G for the Green function) and then S would be the interaction part of the action. Polchinski's equation is usually used in the form (2.4) (or in the form (2.2) for the full action).
In this paper however we use it in the form (2.3). This is a linear equation and is just a free particle Schroedinger equation. The formal solution of this equation in terms of an evolution operator can easily be written down. Writing a formal solution in this form is useful in some situations: the ERG as is usually written down is an infinite number of equations that give the β-function of one coupling parameter in terms of all the other infinite number of coupling parameters. The usual continuum beta function involves only a few of the parameters involving the lower dimensional operators. To go from the first form to the second form one has to solve these infinite number of equations iteratively. [2]. The evolution operator method does this operation in a convenient way (as will be shown). It thus acts as a bridge between the ERG and the continuum field theoretic β-function.

Free theory
Let us understand the connection between the ERG equation and the evolution operator by considering the free theory as a pedagogical exercise. The first step is to construct the field theoretic version of Polchinski's ERG: JHEP07(2017)103

ERG and β-function
The ERG acting on Ψ is: This can be written as an ERG for L. Let us write the Wilson interaction as S ≡ − du L(u, t). We get We could start with a local bare action: where δm 2 (u) = (e 2φ(u) − 1)m 2 is a position dependent coupling(mass), but in general even if we start with a local action, after one iteration of the RG it becomes non-local. So in the present we start with a non-local action Thusṁ The set of β-function equations (2.8) is exact. But the simplicity is a little misleading because z(u, v, t) is a function of two locations u, v and actually represents an infinite number of local (position dependent) coupling functions, which can be defined by Taylor expansions. Note that even for the free field case we get a non local Wilson action.

Evolution operator
Consider the Schroedinger equation: which is solved formally as With t = 1 we get our solution. The solution is known in terms of a kernel So setting t = 1 we get the solution to our original problem: which can also be written in a well known standard form as [5,6,13,28] We can convert the above solution to a field theoretic case and in the free theory, obtain an exact form of the solution to ERG evolution. Working in momentum space, all we need to do is to replace x by X(p). The integral over x ′ becomes a functional integral over X(p) and in the action we need to sum over all p. The "propagator" F (T ) becomes F (p, T ) = G(p, ae T ) − G(p, a): (2.14) In this form it looks a free particle (field) calculation where the propagator is F (p, T ) ≡ G(p, a(T )) − G(p, a) with a(T ) = ae T the moving cutoff. Thus the propagator only propagates the modes that are being integrated out. So, for e.g., when T = 0 it vanishes because no integration has been done.

Free field theory: exact solution of ERG
In the case of the free field the integrations can be carried out exactly.
Thus we have an exact solution for the Wilson action. Furthermore, dz dt = z 2 dF dt = z 2 dG dt which is the second eq. in (2.8). We thus make contact with the differential version of ERG.

Discussion of central charge calculation
We can apply this to the central charge calculation of a free scalar field using the method described in [25,26]. For completeness we review the basic ideas. Later the same ideas will be used for the sine-Gordon theory.

Background
Letĝ αβ = e 2σ δ αβ . As is well known 4 Even though g drops out of kinetic term, we have also seen that the information aboutĝ comes from defining the operator: And what we are calculating is Det − 1 2 ∆. It is thus there in the measure. It is implicit in the above that the UV cutoff is taken to infinity. Thus we can write On the other hand because of scale invariance, we do not have to take Λ → ∞. We can also write without modifying the action, i.e. it is not the Wilson action obtained by integrating out modes from (3.2).
On the other hand in flat space we can set σ = 0 in the above to get: 4 See for instance [14,15].

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Thus we can say that for Λ → ∞, the following statement about integration measures is true: We cannot take finite values of Λ because we may have to integrate over expressions that contain a scale. Now consider adding a mass term: This term explicitly violates scale invariance (or Weyl invariance). We can add a dilaton to make it Weyl invariant: 1 2 d 2 x e 2σ+2φ m 2 X 2 . So if we set δφ = −δσ, it is invariant. Thus the invariance is spontaneously broken rather than explcitly. If we now integrate over X we expect the anomaly to remain the same. Thus we expect Because under the variation δφ = −δσ we get −δσ 1 12π σ which is what one should get. On the other hand because all the modes are frozen -effectively there is no scalar field. The same equations in flat space (σ = 0) give: On the other hand if we evolve from Λ = ∞ to Λ = 0 using the ERG we have to get the same answer. Thus we should get Thus since effectively there is no integration and all degrees are frozen, we must have This gives the expected result ∆c = −1 as you flow from high energies to low energies. There is another way to understand this: we know that This follows from (3.8) with φ replaced by φ + σ. Let us use (3.5) and write Cancelling the (∂σ) 2 term on both sides we reproduce (3.6).

RG evolution and central charge for free scalar
Let us now apply the ERG evolution operator to obtain the φ dependence all along the RG trajectory. This gives us a definition of the c-function.
We start with a non-local action But then we choose z(u, v, 0) = δm 2 (u)δ(u − v) as our bare action at t = 0 and then set δm 2 (u) = (e 2φ(u) − 1)m 2 where φ is the external dilaton field.
The result of the action of the evolution operator is: We are interested in the coefficient of (e 2φ − 1) 2 . Thus we can set X = 0 and evaluate This can be done to get Expanding the logarithm one gets for the quadratic (in φ ) term: Tr[(F m 2 (e 2φ − 1)) 2 ] (3.14) is understood to be evaluated with a cutoff equal to e −t Λ. t → ∞ corresponds to a(t) = ∞. All modes have been integrated out. So the propagator vanishes: G(z, z ′ , ∞) = 0. This is also clear from (5.2). When t = 0 we get the cutoff propagator. We take the cutoff Λ → ∞ which corresponds to a(t) = 0. So we go from the completely unintegrated theory with Λ ≫ m to one with everything integrated out. Thus we get on integrating by parts:

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Substituting in (3.15) we get What we have calculated is −L(u, ∞) + L(u, 0) = −∆L. The change in c is thus -1. The final theory where the scalar field is infinitely massive has c = 0. The initial theory therefore had c = 1. The anomalous transformation under scale changes is provided by the (∂φ) 2 term -this is the argument used by [25,26]. We have obtained it using the ERG.
Note also that (3.14) defines a c-function for any value of t along the flow. It is also clear that it is monotonic.

β-function of φ theory in four dimensions
Now we illustrate the method of calculating the β function for the φ 4 theory using the ERG. We use the Polchinski equation with all the corrections to kinetic term being put into the interactions. Since we are only integrating modes with p > Λ we do not need a mass as a regulator. So we can put m 2 = 0.
The evolution operator is We set What is the action of evolution operator on V : We can keep some terms in the exponent and bring down the rest:

Let us evaluate:
Dφ ′ e x 1 x 2

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Expand the log: In momentum space F can be understood as a propagator with momentum restricted in the range Λ < p < Λ 0 . Thus This is the usual quadratically divergent mass correction. The next term in 1 2 Tr ln[1 − λa 2 φ 2 F ] is: The external momentum is set to zero. This is a correction to λ 4! so we factor out 4! to get: This is the well known β function of the φ 4 theory in four dimensions. What about contributions to β function from λ 3! φφ ′3 ? We calculate using: where J will be set to λ 3! φ 3 in the end. Thus one evaluates All terms necessarily have one factor of the form HJ. To lowest order in λ, H = F . When we set J = λ 3! φ 3 the external momentum is zero (for constant φ) and thus we have an F propagator with zero momentum. This is zero because F is non zero only for momenta greater than Λ. Thus this correction is zero to lowest order.

JHEP07(2017)103 5 Sine-Gordon theory
We now turn to the Sine-Gordon theory. We first derive the β-functions for this theory using the ERG evolution operator. We then calculate the central charge along an RG trajectory.

Green function
The Green function for the Klein Gordon field in two dimensions in Euclidean space is The small t region gets contribution from x 2 = 0 region. This is the UV. A way to regularise this is to cutoff the integral:

Reproducing the continuum β-function
We would like to reproduce the flow for α and b. Since b multiplies φ the latter flow is equivalent to wave function renormalization. So we would like to get terms on the r.h.s. of the ERG involving cos(bφ) or ∂φ∂φ.
The action for the theory is Kinetic term is and we choose α ′ = 2π and substitute it in when we carry out calculations in the later sections.

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The evolution operator acting on the unintegrated theory gives is like a propagator. Also Thus the result is where S n c stands for the connected part of S n and . . . stands for doing the X ′′ integral. This is the cumulant expansion.

Leading order α(t)
Let us bring down one power of (S[X, 0]) c = ( L) c , where the sub-script c signifies that only the connected parts for all terms will be retained, from e −S[X,0] and act on it with the evolution operator . Writing cosine as a sum of exponentials, and noting that the action of the evolution operator gives the same factor for both exponentials, we get: where we have relabelled β → b and X → φ. Powers of a have been added for dimensional consistency. We can use the form given in (5.7) to get is the effect of self contractions in a range of energies (Λ, Λe −t ). This is also the normal ordering factor that one usually obtains which has a(t) replaced by the IR cutoff 1/m. The usual normal ordering integrates out self contractions of all fields,

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i.e up to the IR cutoff. In the ERG only some fields are integrated out and after the ERG evolution the field φ only has lower momentum modes in it, and the pre-factor is the effect of integrating out the rest. One more difference is that normal ordering takes care of only self interactions. The ERG removes all interactions between high momentum modes because the modes themselves are integrated out. This is the origin of terms of the form b 2 F (x 1 , x 2 ) in the exponent. This is like the correlator between two exponentials, but with only some modes -high momentum -participating.
This can be written as . valid for small t. This also gives the leading term in the β-function: where we have approximated α 0 by α(t) to this order in t and α. Thus for ( b 2 4 − 1) > 0 it goes to zero in the infrared and for ( b 2 4 − 1) < 0 it is a relevant variable that goes to infinity in the IR. This is the lowest order K-T flow.
The action of ERG evolution operator on the marginal combination gives

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We are interested in the logarithmically divergent part in order to match with the continuum calculation. (In the above equation one can also replace a by a(t 0 ) and pick terms proportional to ln ( a(t) a(t 0 ) ).) We also take the limit a → 0 so that all powers of a can be set to zero. But in the limit a → 0 there is translation invariance in time (t = ln( a(t) a ))in the evolution equation and as explained in the introduction the beta function cares only about the linear term in t. Furthermore if we assume that a(t) ≈ 1 m , which is the IR cutoff, we can replace y 2 + a(t) 2 by a(t) 2 . Thus we get for the y integral: Putting back the prefactors: Let us take the limit δ → 0 and keep leading terms: If we now take a → 0 we get only the first term. The beta function only cares about the leading logarithm, which is the linear term in t. This is a correction to the kinetic term 1 4π d 2 x 1 ∂ a φ∂ a φ(x 1 ). If we call the correction z(t) we geṫ At this point we can compare with Amit et al. [22]. In their notation β 2 8π = b 2 4 . Thus β 2 8π = δ + 1. This is the same δ that they use. Also α A β 2 = α 4π where α A is the variable used in [22]. Thus we have α = α A 2(δ + 1) Thus we get for the wave function renormalizatioṅ to first order in δ. The zero-eth order term agrees with [22]. The first order term is not universal. The third order piece is calculated in the appendix. Collecting all the beta functions we geṫ

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If we write α = α A 2 , (which is not quite the same as α A 2(1+δ) defined earlier) we geṫ We can compare this with that obtained by Amit et al. (The beta function is given by the flow to the UV and has the opposite sign.) In their notation The non leading term is non universal. It is shown in their paper that B + 2A is universal where A, B are the non leading coefficients. So their value is B + 2A = 5 32 − 2 32 = 3 32 . It can be checked that we get the same ( 2+1 32 ).

Central charge of Sine-Gordon theory
In this section we calculate yet another flow -the c-function defined by Zamolodchikov.
We calculate it using the ERG first. We also compare this with a calculation using a prescription given in [26].

c-function using ERG flow equation
Let us begin with some normalization details: So σ → σ +ξ and φ → φ−ξ is the invariance. a → ae −φ gives the dilaton coupling. Instead of associating the dilaton with a we associate it to the coupling constant α or equivalently to the dimensionful operator cos βX. When we do RG evolution a → ae t ≡ a(t) and as usual and there is no φ associated with it.

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More generally IR cutoff, R, can be varied: R = a(t) = ae t . So we get S = d 2 x a(t) 2 α e −2δ(t+φ(x,t)) : cos βX : To this order Note that the coupling constant has become x-dependent and has to be placed inside the integral: Next order. Now we go to 1 2! S 2 c : We replace α by α(t, φ) as before. This is the scale for the action at the lower energy scale. The propagator has an exponential fall off beyond the IR scale. In our RG a(t) sets the IR scale. So when |y − x| ≈ a(t) the propagator is highly damped. So we are justified in assuming that a < |x − y| < a(t). Thus the total contribution is (letting z = y − x) e −2δ(φ(x)+φ(y)) = 1 − 2δ(φ(x) + φ(y)) + 2δ 2 (φ(x) + φ(y)) 2 + . . .

The relevant part is
Inserting (6.1) for e −2δ(φ(x)+φ(y)) we get for the term in the Wilson action involving φ φ: The integral is log divergent and the divergent piece can be extracted by introducing the regulator a in the limits rather than in the integrand: (The IR end is cutoff anyway by a(t).) Inserting in (6.2) and expanding for small δ we get

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The answer depends on the logarithmic range t. The calculation can be improved if we realise that α is a function of t. We assume that the range of RG evolution t is infinitesimal -dt. Then we keep track of the t dependence of α by writing (6.3) as Noting that dα dt = −2δα we can write dt = − dα 2δα to get When δ > 0 we have an irrelevant operator -α flows to zero under RG evolution. So c(α) > c(0) -which is correct. The change in entanglement entropy is ∆c 3 ln ( l ǫ ) ∆c 3 ln l ǫ = ∆EE = π 2 α 2 δ ln l ǫ If we set α = λ 8π and 2δ = ∆ − 2 we get This has recently been calculated using other methods and also holographically in [27] and the results are in agreement.

Summary and conclusions
In this paper we have studied the RG flow of quantities in field theories. The idea is to use Polchinski's ERG written in terms of an evolution operator. The advantage is that one can directly obtain quantities such as the beta function by looking at the linear dependence on the RG time t. In the limit of cutoff going to infinity this coefficient gives the beta function. This technique was illustrated with a few examples such as the φ 4 theory in four dimensions and also the Sine-Gordon theory in two dimensions -which is the main interest in this paper. We also show that another flow calculation that this method is suited for is that of the c-function. We illustrate it with the case of the free field. We then calculate it for the Sine-Gordon theory. Interestingly if we assume the relation between entanglement entropy and the central charge continues to hold even for the c-function we

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can evaluate the entanglement entropy of the Sine Gordon theory for small values of the perturbation. This has been done using other field theoretic and also holographic methods and there is complete agreement for the lowest order term -which is all that has been calculated [27]. For the Sine-Gordon theory the detailed results of [22] for the solution of the RG equations has been used in this paper to calculate the c-function to higher orders.
There are many open questions. It would be interesting to extend the ideas in this paper to more basic issues in holographic RG and in particular the connection with the RG on the boundary theory. In the context of entanglement entropy it would be interesting to check the match to higher orders. Since the c-function is presumably not a universal quantity (there should be some scheme dependence) at higher orders, these checks have to be made keeping these caveats in mind.
We hope to report on some of these issues soon.
(c) = 2π 2 ln 2 ∆ 2 a 2 + 1 − 2 ln ∆ 2 a 2 + O(a 2 ) (d) = 2π 2 ln 2 ∆ 2 a 2 (e) = −2π 2 ln ∆ 2 a 2 (f)=(e) = −2π 2 ln ∆ 2 a 2 We can now check that the coefficient of ln ∆ 2 ln a 2 is zero. 8π 2 (from I + II) − 4π 2 (from (c)) − 4π 2 (from (d) = 0 The coefficient of ln a 2 is 8π 2 . Thus we get putting back the prefactors 1 8 α 3 (4π) 3 8π 2 2 ln a ∆ Since any value of ∆ is safe for extracting the divergence, we can extend the region of integration to its full value which is α(t) ≈ 1 m . This the modified α 4π . So Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.