###### Abstract

We study the holographic dual of flavors in a Chern-Simons matter theory at non-zero temperature, realized as D6-branes in the type IIA black hole dual in the ABJM background geometry. We consider both massive and massless flavors. The former are treated in the quenched approximation, whereas the massless ones are considered as dynamical objects and their backreaction on the geometry is included in the black hole background. We compute the holographically renormalized action of the probe by imposing several physical conditions. In the limit of massless flavors the free energy and entropy of the probe match non-trivially the first variation of these quantities for the backreacted background when the number of flavors is increased by one unit. We compute several thermodynamical functions for the system and analyze the meson melting phase transition between Minkowski and black hole embeddings.

Thermodynamics of the brane

in Chern-Simons matter theories with flavor

Niko Jokela,^{*}^{*}*niko.jokela@usc.es
Javier Mas,^{†}^{†}†javier.mas@usc.es
Alfonso V. Ramallo,^{‡}^{‡}‡alfonso@fpaxp1.usc.es
and Dimitrios Zoakos^{§}^{§}§dimitrios.zoakos@fc.up.pt

Departamento de Física de Partículas

Universidade de Santiago de Compostela

and

Instituto Galego de Física de Altas Enerxías (IGFAE)

E-15782 Santiago de Compostela, Spain

Centro de Física do Porto

and

Departamento de Física e Astronomia

Faculdade de Ciências da Universidade do Porto

Rua do Campo Alegre 687, 4169-007 Porto, Portugal

###### Contents

- 1 Introduction
- 2 The flavored ABJM background
- 3 D6-brane embeddings at zero temperature
- 4 Flavor brane probes at non-zero temperature
- 5 Minkowski and black hole embeddings
- 6 Brane thermodynamics
- 7 Summary and conclusions
- A Probe action in isotropic coordinates
- B Low temperature (Minkowski embeddings)
- C High temperature limit (black hole embeddings)
- D Mass and condensate
- E Thermal screening
- F Critical embeddings

## 1 Introduction

Recent studies of Chern-Simons matter theories in three dimensions by holographic techniques have provided non-trivial examples of the AdS/CFT correspondence [1, 2] which could be of great help to shed light on the dynamics of some strongly coupled systems in condensed matter physics. The paradigmatic example of these systems is the Aharony-Bergman-Jafferis-Maldacena (ABJM) theory constructed in [3], based on the analysis of [4, 5], where the supersymmetric Chern-Simons matter theories were proposed as the low energy theories of multiple M2-branes.

The ABJM theory is an super Chern-Simons gauge theory in 2+1 dimensions with gauge group with opposite level numbers and . In addition to the two gauge fields, this theory contains two pairs of chiral superfields which transform in the and bifundamental representation. When and are large the theory admits a geometric description in terms of an with fluxes in type IIA supergravity which preserves 24 supersymmetries. The study of this theory and its generalizations has uncovered a very rich structure and has provided new precision tests of the AdS/CFT correspondence (see [6, 7, 8, 9] for reviews of different aspects of the Chern-Simons matter theories).

The ABJM theory can be generalized in several directions. In this paper we will consider the addition of fields transforming in the fundamental representations and of the gauge group. It was proposed in [10, 11] that these flavors can be incorporated in the holographic dual by considering D6-branes that fill the space and wrap an submanifold of the internal space. These configurations are supersymmetric. When the number of flavors is small one can adopt the so-called quenched approximation, in which the flavor D6-branes are considered as probes in the geometry. This approach has been followed in [12, 13, 14, 15].

In [16] a holographic dual of ABJM with unquenched flavor was found by considering a large number of flavor D6-branes which are continuously distributed in the internal space in such a way that supersymmetry is preserved. To find the unquenched solution one has to solve the equations of motion of supergravity with brane sources, which modify the Bianchi identities of the forms and the Einstein equations. If the branes are localized, the sources introduce Dirac -functions in the equations, which makes the problem very difficult to solve. For this reason we will follow the approach initiated in [17] and study the backreaction induced by a smeared continuous distribution of flavor branes. This procedure has been successfully applied to add unquenched flavor in other holographic setups [18, 19, 20] (see [21] for a review and more references). As the smeared flavor branes are not coincident, the flavor symmetry for branes is rather than . Moreover, since we are superimposing branes with different orientations in the internal space, the corresponding supergravity solutions are generically less supersymmetric than the ones with localized flavor. The unquenched solutions with smeared flavors are much simpler than the localized ones and, in many cases the solutions are analytic.

The unquenched solution of type IIA supergravity found in [16] includes the backreaction effects due to massless flavors. The corresponding ten-dimensional geometry is of the form , where is a compact six-dimensional space whose metric is a squashed version of the unflavored Fubini-Study metric of . In this solution the deformation introduced by the flavors is encoded in the squashing factors, which are constant and depend non-linearly on the number of flavors (although the sources of supergravity are linear in ). Notice that the backreacted metric contains an Anti-de Sitter factor. This is related to the fact that the dual Chern-Simons matter theory has conformal fixed points even when the flavors are added (see [22] for a verification of this property in perturbation theory). On the gravity side this conformal behavior is responsible for the regularity of the metric at the IR, contrary to other solutions with unquenched massless flavors [21]. It was checked in [16] that this solution captures rather well many of the effects due to loops of the fundamentals in several observables. In particular, it matches remarkably well with the behavior of the effective number of degrees of freedom of the flavored theory in the Veneziano limit, which was computed in the field theory side using localization in [23].

In sharp contrast to what happens to other flavored backgrounds obtained with the smearing method (see, for example, those of refs. [18, 19, 20]), our supergravity solution has a good UV behavior and, since the metric has an Anti-de Sitter factor, we are dealing with a geometry for which the holographic methods are firmly established and it is possible to apply a whole battery of techniques to perform a clean analysis of the different flavor screening effects. In particular, as it is shown below, it is straightforward to add a further temperature deformation to the flavor deformation and to construct a black hole which contains the effects of massless flavors. This is simply done by including the standard blackening factor in the Anti-de Sitter part of the metric, without modifying the internal space . We can then compute different thermodynamic quantities for this flavored black hole.

When flavor branes are embedded in a black hole geometry the system undergoes a first order phase transition when the branes fall into the horizon [24, 25]. On the field theory side this phase transition corresponds to the melting of mesons in a deconfined plasma. The analysis of the influence of unquenched flavor in this melting transition is clearly a very interesting problem. However, in order to have a complete understanding of this problem in the holographic setup one has to find a black hole solution containing the full backreaction of massive flavors, which is very hard to find. In this paper we will adopt a more modest approach and consider a small number of massive flavors and a large number of massless quarks. The latter will be included in the background, while the massive fundamentals will be treated in the quenched approximation. Accordingly, we will consider a D6-brane probe in the non-zero temperature version of the background found in [16] and we will study its thermodynamic properties, following the same methodology as the one employed in [25] for the D3-D7 and D4-D6 systems.

The action that governs the dynamics of our D6-brane probes contains a contribution from the Dirac-Born-Infeld (DBI) and Wess-Zumino (WZ) terms. This probe action must be renormalized holographically in order to get finite answers for the different thermodynamic functions. At zero temperature one can adopt a gauge for the RR seven-form potential in which the two terms of the action cancel with each other on-shell for the kappa symmetric embeddings of the probe. At non-zero temperature the on-shell action of the probe in this gauge is finite, and the only freedom left by the holographic renormalization is the addition of finite counterterms. These finite terms can be fixed by imposing regularity of at the horizon and by requiring that all the thermodynamic functions for the probe vanish for infinitely massive flavors, as they can be integrated out.

Once the action of the probe is fixed in this way, we should verify that it satisfies a non-trivial compatibility condition with the background. Indeed, let us consider a probe for a massless flavor. In this massless limit the quarks introduced by the probe are of the same type as those of the background. Thus, one can compare the thermodynamic functions of the probe with the variations of these same functions for the background when is increased by one unit. For consistency, these two quantities should be equal. Actually, within the probe approximation one should assume that is large. Then, the variation induced in the background when should be computed by a Taylor expansion in which only the first term is kept. We will verify that this compatibility condition is indeed satisfied in our case, which is a highly non-trivial test because the dependence of the background on is non-linear. After passing successfully this test, we are ready to study systematically the thermodynamics of the probe brane. In general, the main objective is to determine the dependence of the different observables on the number of flavors of the background, as well as the departure from conformality induced on the system by the probe.

The plan of the rest of this paper is the following. In Section 2 we will present our flavored black hole background and compute some of its thermodynamic functions. In Section 3 we will analyze the flavor brane embeddings at zero temperature and extract some useful information which will be needed in the black hole case. In Section 4 we will study the action of the probe in the non-zero temperature geometry and we will check that the compatibility condition mentioned above is satisfied. In Section 5 we shall study in detail the two types of embeddings, Minkowski and black hole, and we shall analyze the first order phase transition between them. Section 6 is devoted to the calculation of the different thermodynamic functions of the probe (free energy, internal energy, entropy, and normal speed of sound). Section 7 contains a summary of our results and a discussion. The paper is completed with several appendices, which contain some explicit calculations and details not included in the main text.

## 2 The flavored ABJM background

In this section we will present the non-zero temperature version of the deformed ABJM background found in [16]. The ten-dimensional metric, in string frame, of this supergravity solution takes the form

(2.1) |

where is the radius of curvature and is the metric of a black hole in the four-dimensional Anti-de Sitter space, given by

(2.2) |

and is the metric of the compact internal six-dimensional manifold.^{1}^{1}1Unless otherwise stated, we will use units for which .
In (2.2) the blackening factor is given by

(2.3) |

where the horizon radius is related to the temperature by

(2.4) |

The internal metric in (2.1) is a deformed version of the Fubini-Study metric of . This deformation is due to the backreaction of the massless flavors, generated by the D6-branes, and can be simply stated when the manifold is represented as an -bundle over , with the fibration constructed by using the self-dual instanton on the four-sphere. Explicitly, this metric can be written as

(2.5) |

where and are constant squashing factors, is the standard metric for the unit four-sphere, () are Cartesian coordinates that parameterize the unit two-sphere () and are the components of the non-Abelian one-form connection corresponding to the instanton.

The squashing factors and in (2.5) encode the effect of the massless flavors in the backreacted metric. Indeed, when the metric (2.5) is just the canonical Fubini-Study metric of a manifold with radius and (2.1) is the metric of the unflavored ABJM model at non-zero temperature. The parameter represents the relative squashing of the part of the metric with respect to the part due to the flavor, while parameterizes an internal deformation which preserves the - split of the twistor representation of . The explicit expression for the coefficients and of the smeared solution of [16] is given below. They depend on the number of colors and flavors , as well as on the ’t Hooft coupling , through the combination

(2.6) |

where the factor is introduced for convenience. The radius can be also expressed in terms of and the deformation parameter (2.6) (see eqs. (2.29) and (2.30)).

The type IIA supergravity solution found in [16] contains, in addition to the metric (2.1), a constant dilaton and RR two- and four-forms and . In order to specify the form of the latter, let us introduce a specific system of coordinates to represent the metric (2.5). First of all, let () be the left-invariant one-forms which satisfy . Together with a new coordinate , the ’s can be used to parameterize the metric of a four-sphere as

(2.7) |

where is a non-compact coordinate. The instanton one-forms can be written in these coordinates as

(2.8) |

Let us next parameterize the coordinates of the by two angles and (, ), namely

(2.9) |

Then, one can easily prove that

(2.10) |

where and are the following one-forms:

(2.11) | |||

(2.12) |

Using these results we can represent the ten-dimensional metric (2.1) as

(2.13) |

We shall next consider a rotated version of the forms by the two angles and . Accordingly, we define three new one-forms :

(2.14) |

In terms of the forms defined in (2.14) the line element of the four sphere is obtained by substituting in (2.7). Let us next define the one-forms and ,

(2.15) |

in terms of which the metric of the four-sphere is

(2.16) |

With these definitions, the ansatz for for the flavored background written in eq. (5.6) of ref. [16] is

(2.17) |

where is a constant squashing parameter between the and components of (2.17). In the unflavored ABJM solution of [3] the is given by (2.17) with . For a general value of the two-form is not closed. Indeed, one can easily verify that

(2.18) |

where is the following three-form

(2.19) |

Thus, when the Bianchi identity for is violated. This violation is due to the presence of a delocalized set of D6-branes, whose Wess-Zumino action can be written as

(2.20) |

where is the RR seven-form potential and is a charge distribution three-form. Clearly, the term (2.20) induces a source for , which modifies the Maxwell equation of . Taking into account that , one easily concludes that the equation of motion for just takes the form of the modified Bianchi identity (2.18). Thus, one identifies the three-form written in (2.19) with the one parametrizing the distribution of the smeared set of D6-branes. Actually, from this identification one can relate the constant to the total number of flavors . Indeed, one gets [16] the simple equation:

(2.21) |

It is obvious from (2.21) that is simply related to the deformation parameter introduced in (2.6),

(2.22) |

In the solution of [16] the squashing parameters and are related by a quadratic equation, which is obtained by requiring that the background preserves supersymmetry at zero temperature. This quadratic equation is

(2.23) |

By solving this equation for and using (2.21) one can obtain as a function of the deformation parameter ,

(2.24) |

Moreover, the solution of the BPS system of [16] allows to relate the parameter to the squashing factors and :

(2.25) |

From this equation we get the explicit expression of in terms of the deformation parameter :

(2.26) |

By construction when , whereas in the flavored solutions these coefficients are greater than one. In order to have a better idea of the behavior of and it is quite useful to expand them in powers of . We get

(2.27) |

Notice, however, that and reach a finite limiting value when the deformation parameter is very large. Indeed, one can check from (2.24) and (2.26) that

(2.28) |

To fix completely the metric (2.1) we need to know the value of the radius . In the unflavored case is proportional to the square root of the ’t Hooft coupling . This value gets deformed by the backreaction of the flavors. Actually, we have [16],

(2.29) |

where is defined as the following function of the deformation parameter:

(2.30) |

It was shown in [16] that characterizes the corrections of the static quark-antiquark potential due to the screening produced by the flavors. In Fig. 1 we depict and (2.36) as functions of the deformation parameter .

The solution is completed by a constant dilaton given by

(2.31) |

and a RR four-form , whose expression is

(2.32) |

where is the volume-form of the four-dimensional black hole (2.2). The regime of validity of the type IIA supergravity description can be obtained by requiring that and . For the flavored ABJM background at zero temperature these two conditions were worked out in detail in [16] and will not be discussed further here.

In the zero temperature case this background was found in [16] by solving the system of first order BPS equations required to preserve supersymmetry. Then, one can verify that the solution satisfies the second order equations of type IIA supergravity with sources (see appendix D of [16]). In the black hole case one can easily check that these equations of motion are still satisfied after the introduction of the blackening factor in the metric.

### 2.1 Thermodynamics of the background

Let us now find the values of the different thermodynamic functions for the flavored black hole presented above. We begin by computing the entropy density , which is given by:^{2}^{2}2We use the same conventions as the first paper in [20].

(2.33) |

where is the volume at the horizon of the eight-dimensional part of the space obtained by setting in the ten-dimensional geometry and is the infinite volume of the 2d space directions . The volume has to be computed with the Einstein frame metric, which in our case is obtained by changing by in (2.1) and (2.5). After a simple calculation one can check that is given by

(2.34) |

We can now use the values of the different factors appearing on the right-hand side of (2.34) to obtain the value of the entropy density in terms of gauge theory quantities. Taking into account that, in our units, , we get

(2.35) |

where

(2.36) |

The quadratic dependence of the entropy with the temperature is a reflection of the conformality of the system which, in our solution, is not affected by the massless flavors. Notice that displays the characteristic behavior of the effective number of degrees of freedom of the ABJM theory in the ’t Hooft limit. The correction to this behavior introduced by the flavors is parameterized by the function , which was introduced in [16] and shown to be very close to the function obtained by using the localization technique. The function determines how the volume of the internal manifold (and, hence, the area of the horizon) changes due to the addition of flavor.

The internal energy density can be obtained from the ADM energy,

(2.37) |

In (2.37) is the Einstein frame metric of the hypersurface. The integral is taken over this hypersurface for a large value of the radial coordinate. The symbols and denote the extrinsic curvatures of the eight-dimensional subspace within the nine-dimensional (constant time) space, at finite and zero temperature, respectively. For an arbitrary hypersurface is given by

(2.38) |

with being a normalized vector perpendicular to the surface. For a constant hypersurface,

(2.39) |

and one can show that for our background becomes

(2.40) |

By using these results it is easy to find the value of the integrand in (2.37),

(2.41) |

where is the internal metric (2.1). It is now immediate to obtain the internal energy density of the flavored black hole,

(2.42) |

Again, the dependence on the temperature is just the one expected for a conformal system and the flavor dependence is determined by the function . Moreover, the free energy density can be obtained from the thermodynamic relation , yielding,

(2.43) |

As a consistency check we notice that , as it should. It is also worth pointing out that the free energy density can be computed directly from the regularized Euclidean action (see the first paper in [20] for a similar calculation for the D3-D7 black hole). The regularization is performed by subtracting the action at zero temperature with the Euclidean time suitably rescaled. Furthermore, in the action one must include the standard Gibbons-Hawking surface term. The final result of this calculation, which will not be detailed here, is just the same as in (2.43).

## 3 D6-brane embeddings at zero temperature

One key objective of this paper is to study the properties of flavor brane probes embedded in the flavored black hole background described in Section 2. Before dealing with this problem in full generality, let us analyze the case in which the temperature of the background is zero, which corresponds to taking the blackening factor equal to one in the formulas of Section 2.

The kappa symmetric embeddings of the flavor D6-branes that preserve the supersymmetry of the zero temperature background were studied in [16]. As argued in [10], these D6-branes should extend along the three Minkowski directions , the radial coordinate , and wrap a three-dimensional submanifold of the compact internal space. For large values of the radial coordinate the metric of this three-dimensional submanifold should approach that of a (squashed) . In our representation, it was shown in [16] that this internal submanifold is obtained by extending the D6-brane along the base in such a way that the pullback of the one-forms and vanish. Accordingly, let us consider a configuration such that , where the hat denotes the pullback to the D6-brane worldvolume. Moreover, for the pullback of we just take , where is an angular coordinate. We will also assume that the brane is extended along the coordinate of the fiber and that the other coordinate is a function of the radial coordinate , . Therefore, we will choose the following set of worldvolume coordinates

(3.1) |

Then, the induced metric (at zero temperature) on the D6-brane worldvolume becomes

(3.2) |

where is the following three-dimensional metric

(3.3) |

If we redefine the angular coordinates as

(3.4) |

then the 3d metric becomes

(3.5) |

where is assumed to be a function of . The range of the angular coordinates in (3.5) is,

(3.6) |

Notice that, in these coordinates, the massless configurations whose backreaction is included in the background of Section 2, correspond to embeddings with being constant and equal to . In order to simplify the study of all possible embeddings that satisfy the equations of motion of the probe, it is convenient to choose an isotropic system of coordinates. To find these coordinates, let us consider the part of the induced metric (3.2), which can written as,

(3.7) |

We want to find a new radial coordinate such that the first term inside the brackets in (3.7) becomes and the whole right-hand side of (3.7) is proportional to . Clearly, we must require

(3.8) |

and thus (3.7) becomes

(3.9) |

Eq. (3.8) can be immediately integrated, with the result

(3.10) |

Notice that the change of the radial coordinate is only non-trivial in the flavored case with . In terms of this variable, the ten-dimensional metric (2.1) (for ), becomes

(3.11) |

where is the metric (2.5) of the squashed .

Let us now introduce a system of Cartesian-like coordinates , defined as

(3.12) |

The inverse relation is

(3.13) |

and, since , the line element (3.7) becomes

(3.14) |

Let us now consider embeddings of the D6-brane in which . Then, the induced metric takes the form

(3.15) |

with . The embeddings corresponding to massless flavors are the ones for which . In the general case, the determinant of the induced metric takes the form

(3.16) |

In order to obtain the explicit form of the embeddings, let us now study the action of the probe brane. We begin by computing the DBI action, which is given by

(3.17) |

where the tension of the D6-brane in our units. Let us use (3.16) in (3.17) and integrate over the angular coordinates , , and . We define a Lagrangian density as

(3.18) |

where

(3.19) |

with being the following constant

(3.20) |

Next, let us compute the WZ term of the action, which becomes

(3.21) |

where is the RR seven-form potential () and, as before, the hat denotes the pullback to the worldvolume. In this zero temperature case the RR seven-form potential is naturally given in terms of the calibration seven-form that characterizes the G-structure of the supersymmetric solution. Indeed, we can take as

(3.22) |

The seven-form is naturally defined in terms of a fermion bilinear which, in turn, can be obtained from the projections satisfied by the Killing spinors of the background. This calculation was performed in [16] and here we will limit ourselves to recall this result. As shown in [16], to represent it is useful to define the following basis of one-forms:

(3.23) |

which are a frame basis for the zero-temperature version of the metric (2.1). In terms of the forms (3.23) the form can be written as [16],

(3.24) |

To evaluate the WZ action we need to compute the pullback of to the worldvolume. Let us write the pullbacks of the frame one-forms (3.23) in the coordinates. In this calculation it is convenient to use

(3.25) |

We find

(3.26) |

By inspecting these pullbacks one readily verifies that the only non-zero contributions to are

(3.27) |

Thus, after integrating over the angular variables, we can write

(3.28) |

with the Lagrangian density

(3.29) |

Therefore, the total Lagrangian density is

(3.30) |

Clearly, is a solution of the equations of motion derived from (notice that the on-shell action for this configuration vanishes). This is just the kappa symmetric solution that preserves SUSY which was found in [16].^{3}^{3}3Notice that, in the angular parameterization of [16], the solution reads , where . Let us now study the form of a general solution in the UV region of large . In this case one can approximate in (3.30) and take small. At second order in , we find that can be approximately taken as

(3.31) |

The equation of motion derived from this second-order Lagrangian is simply

(3.32) |

and can be integrated trivially

(3.33) |

In (3.33) and are constants, which should be related to the mass of the quarks and to the vacuum expectation value of the corresponding bilinear operator (see below), respectively. The power of of the subleading term in (3.33) should determine the conformal dimension of the bilinear operator. Indeed, let us consider a canonically normalized field in with conformal dimension . The behavior of near the boundary of is

(3.34) |

where (the boundary value of ) is identified with the source of the dual gauge theory operator and the coefficient is identified with its VEV. In (3.34) is the dimension of and is the canonical coordinate of (in terms of which the metric takes the form ). It is clear that this canonical coordinate is just the one in (2.2). In the UV, and are related as , and therefore we can rewrite (3.33) in terms of as

(3.35) |

In order to extract the dimension of the operator dual to the scalar , let us rewrite (3.34) in such a way that the asymptotic value of the right-hand side is constant,

(3.36) |

Clearly, by comparing (3.36) and (3.35) we find that, in our flavored ABJM case, , which yields

(3.37) |

in agreement with the value obtained in [16] for the dimension of the bilinear operator . Notice also that is the dimension of the source (the mass in our case). This dimension is just in the flavored ABJM case. Thus, the mass anomalous dimension is

(3.38) |

It is evident from (3.38) that the anomalous dimension depends on the number of flavors and, according to (2.28), it becomes maximum when :

(3.39) |

As it was already mentioned, the asymptotic value should be related to the quark mass . To find the precise relation let us consider a fundamental string extended from the origin to the point with at . The induced metric on the worldsheet of this string is

(3.40) |

whose determinant is

(3.41) |

The Nambu-Goto action for this string is

(3.42) |

The action per unit time should be identified with . Thus, by using (2.29) we arrive at

(3.43) |

where is the ’t Hooft coupling and is the function of that has been defined in (2.30). We have included a factor of to reinstate the correct dimensions.

The constant in (3.35) should be related to the vacuum expectation value of the meson operator (the quark condensate). In order to find this relationship we should relate to the derivative of the action with respect to the mass parameter . In principle, to perform this calculation we should holographically renormalize the action to ensure its finiteness [26, 27]. It turns out, however, that the action corresponding to the Lagrangian density (3.30) is convergent and, therefore, this renormalization is not needed. Indeed, by using the asymptotic behavior (3.35) we obtain for large ,

(3.44) |

and, since the maximum value of is , the integral over is convergent as claimed. Notice that this convergent behavior is a consequence of the particular gauge for chosen. Indeed, performing a gauge transformation of the type is equivalent to adding a boundary term to the action of the probe and to choose a particular renormalization scheme. In our gauge is chosen to be the calibration form and, as a consequence, the action for a supersymmetric embedding vanishes. For a more general embedding the WZ term introduces a subtraction of the DBI term, which renders the total action finite.

The probe configuration is obtained by solving the equation of motion derived from the Lagrangian density (3.30) for . In this process we have to impose boundary conditions at some value of the coordinate. The simplest thing is to take as this initial value of the coordinate and to integrate the system outwards. It is easy to verify from the limit of the differential equation at that the only possibility to have non-singular solutions is to take and as initial conditions. At the UV region of large the function must behave as in (3.33), where the constants and are not independent since both should be determined by the IR value of . The on-shell action is obtained by evaluating the integral of for these configurations. It can be considered as a function of the mass parameter . The derivative of with respect to can be computed as follows: