Contents

Plasma photoemission from string theory


Babiker Hassanain111babiker@thphys.ox.ac.uk and Martin Schvellinger222martin@fisica.unlp.edu.ar


The Rudolf Peierls Centre for Theoretical Physics,

Department of Physics, University of Oxford.

1 Keble Road, Oxford, OX1 3NP, UK.


IFLP-CCT-La Plata, CONICET and

Departamento de Física, Universidad Nacional de La Plata.

Calle 49 y 115, C.C. 67, (1900) La Plata,

Buenos Aires, Argentina.

Abstract

Leading ’t Hooft coupling corrections to the photoemission rate of the planar limit of a strongly-coupled SYM plasma are investigated using the gauge/string duality. We consider the full type IIB string theory corrections to the supergravity action, including higher order terms with the Ramond-Ramond five-form field strength. We extend our previous results presented in [1]. Photoemission rates depend on the ’t Hooft coupling, and their curves suggest an interpolating behaviour from strong towards weak coupling regimes. Their slopes at zero light-like momentum give the electrical conductivity as a function of the ’t Hooft coupling, in full agreement with our previous results of [2]. Furthermore, we also study the effect of corrections beyond the large limit.

1 Introduction

Electrically charged particles in a quark gluon plasma (QGP) emit photons. An analysis of these photons can lead to very valuable information about the medium in which they are produced. On the one hand, transport coefficients related to the electric charge such as electrical conductivity and charge diffusion constant characterize the dynamics of long wavelength, low frequency fluctuations in a plasma. They are effectively related to ultra-soft photons, i.e. those with momentum much smaller than the equilibrium temperature of the medium, . Ultra-soft photons eventually probe the hydrodynamical regime of the plasma, with momentum , where is the ’t Hooft coupling defined as , where is the SYM theory coupling and the rank of its gauge group, in the present case. On the other hand, it is possible to scrutinize a thermally equilibrated plasma for a long range of emitted photon wavelengths. This precisely gives shape to the photoemission rate from a plasma as a function of the energy of the photons. It includes ultra-soft, soft and hard photons, thus providing extremely useful information about the dynamical structure of the medium.

For a weakly coupled QCD plasma, transport coefficients and photoemission rates have been calculated using perturbative quantum field theory in [3, 4, 5, 6, 7, 8] and references therein. These references are particularly important since Arnold, Moore and Yaffe have obtained the first complete leading order results for the photoemission rates in QCD [5]. They conclude that, in addition to well known particle processes, near-collinear Bremsstrahlung and inelastic pair annihilation also make leading order contributions. The Landau-Pomeranchuk-Migdal (LPM) suppression, which is the effect produced by multiple soft scatterings, may occur during the emission of the photon and has important implications on the consistent treatment of the above mentioned processes. The LPM effect leads to an suppression of these near-collinear processes.

There are indications, however, that the QGPs produced at the Relativistic Heavy Ion Collider (RHIC) and at the Large Hadron Collider (LHC) are in the strongly-coupled regime of QCD [9, 10, 11, 12, 13, 14, 15, 16, 17]. This is where the gauge/string duality enters. This duality allows us to compute properties of a strongly coupled gauge theory in terms of a weakly coupled holographic dual string theory description [18, 19, 20]. We ought to admit that at present there is no complete or exact holographic string theory dual model which accounts for all the relevant properties of real QCD, not even in the planar limit of the gauge theory. For reasons which shall be explained below, the holographic string theory dual model which has been considered so far for these investigations is in fact dual to the large limit of the strongly-coupled supersymmetric Yang-Mills (SYM) plasma.

The holographic dual model of the planar limit of the strongly-coupled SYM plasma is defined in terms of a type IIB supergravity background given by a direct product of an Anti-de Sitter-Schwarzschild black hole in five dimensions () times a five sphere . There is a number of considerations to take into account at the moment of extrapolating this dual description of the large limit of SYM theory in order to make contact with QCD. Firstly, as it is well known at zero temperature these theories are very different. Indeed, the field content is different: while QCD has three colour degrees of freedom and three flavours, matter is in the fundamental representation of the gauge group , it shows colour confinement, has explicit and spontaneous chiral symmetry breaking, and displays a discrete spectrum; on the other hand, in the SYM theory all their fields transform in the adjoint representation of , it is not a confining theory, conformal symmetry is preserved at quantum level, and it is a supersymmetric theory with the maximal number of supersymmetries in four dimensions. At finite equilibrium temperature, , above the critical temperature of QCD, , where hadrons become a deconfined QGP, there are two regimes. For , again the two types of plasmas related to these two theories behave very differently too: while in QCD the coupling runs to weak coupling, leading to a free gas of quarks and gluons; in the case of the SYM plasma, the coupling, which remains constant, is strong. Thus, it leads to a strongly coupled plasma. However, in the intermediate region where is just above , both plasmas behave somewhat similarly. In this case QCD behaves as a strongly coupled plasma of gluons and fundamental matter. These degrees of freedom are deconfined, there is screening and the correlation lengths are finite. Interestingly, the SYM plasma shares those properties because it is a strongly coupled plasma of gluons and adjoint matter fields, it is also deconfined, shows screening, and has finite correlation lengths. Moreover, quantum field theories lattice calculations indicate that for certain properties the similarities can be made even quantitatively (see for instance [21] and references therein). Therefore, one may assume that for but not , there is a parametric region where one can focus on in order to describe the rates for the emission of photons from a thermally equilibrated SYM plasma using the gauge/string duality at finite yet strong ’t Hooft coupling.

A very important step towards the understanding of the photoemission process and electric charge transport coefficients of QGP in terms of the SYM plasma has been done in a very nice paper by Caron-Huot, Kovtun, Moore, Starinets and Yaffe [22]. They consider the two limiting situations: for very large and very small ’t Hooft coupling. In the strongly coupled case they consider the pure type IIB supergravity description of the large limit of the SYM plasma, which we summarize in section 3. In the opposite limit they consider a perturbative quantum field theory description of the SYM plasma, using similar ideas as in [5]. In section 5 we shall briefly review some perturbative results of [22].

In the light of the new experimental findings suggesting that the QGP plasma at RHIC and LHC is in the strongly coupled regime of QCD, a more realistic outlook requires a consideration of the ’t Hooft coupling expansion around the infinitely strongly-coupled regime of the plasma. On the string theory side, we must therefore consider the full type IIB string theory corrections to the supergravity action. It includes a number of terms which arise from the supersymmetric completion to the standard power-four ten-dimensional Weyl-tensor. These new terms are constructed from a rank-6 tensor which contains the Ramond-Ramond five-form field strength. This is indeed a very complicated task from the technical point of view. However, it is worth carrying out since it yields the precise structure of the ’t Hooft coupling corrections to the strong coupling regime. Using this procedure we have obtained very interesting results, which we briefly describe here and present in full detail in section 4.

Our results show the following features. Firstly, the slopes of the photoemission rates, which at zero light-like momentum give the electrical conductivity as a function of the ’t Hooft coupling, are in full agreement with our previous results of [2]: the electrical conductivity increases as the ’t Hooft coupling decreases. This concerns the hydrodynamic regime of the plasma. Secondly, for higher momentum, the height of the peaks decrease as the ’t Hooft coupling increases (i.e. as we approach the limit of infinite coupling), their maxima are shifted towards the ultraviolet and the photoemission rate curves cross downwards the limiting strongly coupled curve for momentum around 3 times the equilibrium temperature. Another important feature which comes from our results is that the number of emitted photons increases as the ’t Hooft coupling weakens. These features show an interpolating trend from the supergravity calculation of the strongly coupled gauge theory towards the perturbative quantum field theory calculation in the weakly coupled supersymmetric Yang-Mills plasma. In addition to describing these effects in more detail in the general discussion and conclusions section below, we will also consider the effect of including non-planar perturbative corrections from higher derivative terms in the type IIB action as well as non-perturbative contributions due to D-instanton effects.

In section 2 we briefly describe generalities about the formalism behind the calculation of plasma photoemission rates based on the computation of two-point correlators of electromagnetic currents. A review of strongly coupled SYM plasma results entirely obtained within pure type IIB supergravity, i.e. with no string theory corrections, is presented in section 3. In section 4, which is the longest section of the paper, we introduce details of the formalism and results from our calculations of the leading ’t Hooft coupling corrections to the photoemission rate of a strongly-coupled SYM plasma using the gauge/string duality. Section 5 is devoted to a very brief review of results in the weakly coupled regime. The material of this section is used in the last section of the article in order to carry out a general discussion of our results.

2 Derivation of photoemission rates in SYM plasmas

In this section we very briefly review the formalism needed in order to derive the photoemission rate in plasmas from thermal field theory. Since we expect to be able to compare our results with those of reference [22], when possible we mainly follow its notation through this paper. Also, the same assumptions as in [22] are considered here: the plasma is in thermal equilibrium; we do not include prompt photons produced by the initial scattering of partons from the colliding nuclei; and, the electromagnetic coupling constant, , is considered small enough in order to ensure that photons are not to be re-scattered.

Consider the minimal coupling of a photon to the electromagnetic current of the SYM plasma. Recall that the gauge supermultiplet is , where is the colour index, , , and all the fields transform in the adjoint representation of the gauge group. They are gauge bosons, 4 Weyl fermions and 6 real scalars, respectively. Furthermore, since there is an anomaly free global -symmetry, there is an associated global -symmetry current, . The way to consider the electromagnetic coupling is by adding a gauge field which couples to the conserved current of a subgroup of the full -symmetry group [22], under the assumption that, to leading order in , . Thus, the Lagrangian can be written as

(1)

where is the Lagrangian of the SYM theory and is the kinetic term of the photon field.

We denote the photon four-momentum as , which is a null four-vector having its time component fixed by the on-shell condition . We use the mostly plus signature for the Minkowski metric in four dimensions, denoted by . First, let us consider the Wightman function of electromagnetic currents defined as

(2)

which in thermal equilibrium is related to the spectral density by

(3)

with the Bose-Einstein distribution function . In addition, the spectral density is given by the imaginary part of the retarded current-current correlation function

(4)

The number of photons which are produced per unit time per unit volume is denoted by . At leading order in the photoemission rate is given by

(5)

Notice that this formula for the photoemission rate holds to leading order in the electromagnetic coupling . On the other hand, and very importantly for our purposes, it is valid non-perturbatively in all other interactions, i.e. the strong interaction [22].

It is worth mentioning that the slope of the photoemission rate in the zero-frequency limit is proportional to the electrical conductivity of the plasma, , which can also be determined by the current-current correlator using the Kubo formula:

(6)

In the next section we describe the computation of the plasma photoemission rate in infinitely strongly-coupled plasma.

3 Review of photoemission rates at strong ’t Hooft coupling

The background, which is an exact solution of type IIB supergravity, is given by

(7)

where , and is the radius of the and the five-sphere. The -boundary is at and the black hole horizon is at . For the coordinates we use indices . It is well known that this is the holographic dual background to the large limit of the SYM theory at finite temperature .

As mentioned, the purpose of the present work is to investigate the ’t Hooft coupling corrections to the photoemission rate of a SYM plasma produced by the leading order corrections to the pure type IIB supergravity calculation. In this section we briefly review some of the calculations of [22], which are applicable for the limit. The idea is to obtain the correlation functions of two -symmetry currents using the methods developed in references [23, 24].

The general form of the correlator at finite temperature is obtained by taking into account rotation and gauge invariance:

(8)

where the transverse and longitudinal projectors are defined such that , , and , with . We use the notation for the photon light-like momentum defined in the previous section and . The trace of the spectral function is

(9)

For light-like momentum only contributes. Therefore, it is the only relevant part for the computation of the photoemission rate.

The gauge/string duality establishes a precise prescription to compute a two-point correlator of conserved currents in a strongly coupled SYM theory. The idea is the following: the insertion of an operator of the SYM theory at the -boundary induces a fluctuation of a certain ten-dimensional background field. Specifically, using the gauge/string duality prescription, a global symmetry current in the SYM theory couples to a gauge field in the bulk, . From the SYM theory point of view the group is a subgroup of the -symmetry group of the SYM theory. Recall that the group is isomorphic to the group, which obviously is the global symmetry which generates rotations among the 6 real scalars of the vector supermultiplet of the gauge theory. On the other hand, from the supergravity side, the isometry group of the five sphere is . Thus, there is a subgroup, which is related to vector fluctuations of the metric, whose gauge field is precisely the Abelian gauge field. Therefore, the point is to solve the linearised equations of motion for the vector perturbations of the metric. The definition of the two-form field strength is . With the identification one can write down the EOMs for the vector fluctuation by splitting them into the transverse , and longitudinal components as follows:

(10)
(11)

where primes denote derivatives with respect to the radial coordinate , and one defines and . The solution of these EOMs have been discussed in [22], so here we just quote their results in the following equations. First, notice that the correlators are determined by the boundary term of the five-dimensional on-shell Maxwell action

(12)

and by applying the Lorentzian AdS/CFT prescription [23] it turns out that the transverse component which is the only one actually needed for the computation of the photoemission rate is given by [22]

(13)

For light-like momenta there is an analytical solution to the EOM above which can be written in terms of a hypergeometric function

(14)

Thus, the trace of the spectral function for light-like momenta is

(15)

In addition, the electrical conductivity is given by

(16)

which has been obtained from the Kubo formula quoted in the previous section.

Finally, the photoemission rate is given by

(17)

which holds in the large limit and for large (where the supergravity approximation is valid, ), and is valid for the whole range of photon energies.

Now, we proceed to investigate the leading ’t Hooft coupling corrections to these expressions and analyse their physical implications.

4 ’t Hooft coupling corrections to photoemission rates

In this section we present the general corrections to type IIB supergravity action at leading order in . Firstly, in subsection 4.1 we describe the formalism needed to account for higher derivative corrections to the effective IIB action. Then, we focus on string theory corrections and develop the vector perturbations we need for the computation of the current-current correlators. In subsection 4.2 we carry out the computation of ’t Hooft coupling corrections to photoemission rates, whose results we show in subsection 4.3. Our results concerning the effects of leading corrections and non-perturbative instanton contributions are restricted to the electrical conductivity of the plasma, and are presented in the discussion and conclusions, in the last section of the paper.

4.1 Higher derivative corrections to the effective IIB action and vector perturbations

To begin with, we consider the leading type IIB string theory corrections to the supergravity action which are given in the term . The total action that we shall consider is

(18)

At the strong ’t Hooft coupling limit the holographic dual model is derived from type IIB supergravity, i.e. for . This contains the Einstein-Hilbert action coupled to the dilaton and the Ramond-Ramond five-form field strength

(19)

Effects of higher curvature terms which includes , perturbative corrections as well as instanton corrections were considered in the presence of D3-branes in type IIB string theory by Green and Stahn in [25]. This reference proposes a supersymmetric completion of the term, where is the ten-dimensional Weyl tensor, leading to the following correction:

(20)

where is the complex scalar field given by , with being the axion and the string coupling. The function is the so-called modular form. The tensor tensor is defined in terms of the Weyl tensor and

(21)

as given in [25, 26, 27, 28]

(22)

The action (20) was arrived at using the fact that the physical content of type IIB supergravity can be arranged in a scalar superfield , where , with , is a complex Weyl spinor of . The matrices have the usual definitions [27].

The modular form is presented in [29] and is given by the following expression

(23)

where is the modified Bessel function of second kind which comes from the non-perturbative D-instantons contributions. Recall that the zeta function is the coefficient of the first perturbative correction in the Eisenstein series of the modular form. Note that in the background we consider with coincident parallel D3 branes, the axion vanishes, thus , while . Therefore, for small values of the modular form becomes

(24)

It is interesting to mention that Green and Stahn also have shown that the D3-brane solution in supergravity does not get renormalised by higher derivative terms [25]. Previously Banks and Green had shown that is a solution to all orders in [30].

Now, we focus on the large limit of the dual SYM theory. Later on, in the conclusions, we shall return to the consequences of the general corrections to the electrical conductivity.

The finite leading ’t Hooft coupling corrections are accounted for by the following action [28]

(25)

obtained from the action (20) in the large limit, where , where . Since , we get . This action was computed in [27], using the methods of [31].

The term is a dimension-eight operator, defined as follows:

(26)

where is the Weyl tensor. The tensor is defined by

(27)

where the indices and are antisymmetrized in each squared brackets, and symmetrized with respect to interchange of [27].

At finite temperature the metric only gets corrections from the term. This is so because the tensor vanishes on the uncorrected supergravity solution [28]. The solution to the Einstein equations derived from the pure supergravity action (19) is an background. There are units of flux of through the sphere, and the volume form of is denoted by . On the field theory side, is the rank of the gauge group, and it corresponds to the number of parallel D3-branes whose back-reaction deforms the space-time leading to the above metric in the near horizon limit. The current operator is dual to the -wave mode of the vectorial fluctuation on this background.

Next, we have to obtain the Lagrangian for the vectorial perturbation in this background. Thus, we must construct a consistent perturbed Ansatz for both the metric and the Ramond-Ramond five-form field strength, such that a subgroup of the -symmetry group is obtained [32, 33, 34]. Then, by plugging this consistent perturbation Ansatz into the full action (up to ) and integrating out the five-sphere, one obtains the desired action for the gauge field in the . Therefore, by studying the bulk solutions of the Maxwell equations in the with certain boundary conditions we can obtain the retarded correlation functions [23, 24, 22] of the operator .

Higher-curvature corrections to the type IIB supergravity action correspond to finite ’t Hooft coupling corrections in the field theory. Suppose that we are interested in a certain observable of the gauge theory, . If one carries out a series expansion of it in inverse powers of the ’t Hooft coupling one schematically can write it as: . The power is a positive number corresponding to the leading correction to the type IIB supergravity action. In the present case, we consider that is the product of two electromagnetic currents. Thus, we obtain the leading correction in using the gauge/string duality. The leading order corrections come from terms in the ten-dimensional action. It is important to recall that these corrections dot no modify the metric at zero temperature [30]. At finite temperature things are different as shown in [35, 36] where corrections to the metric were obtained, and then further improved in [26, 37, 38].

Higher curvature corrections on the spin-2 sector of the fluctuations have been investigated in [39, 40, 41, 42], among other references. They are relevant to the computation of the viscosity and mass-diffusion constants of the plasma.

In our case, we investigate vector fluctuations of the background. The method to carry out the calculation consists of two steps. Firstly, we have to obtain the minimal gauge-field kinetic term using the vector-perturbed metric including the corrections to it, and the same for the five-form field strength. Then, the corrections to the gauge field Lagrangian coming directly from the higher-derivative operators have to be computed. The reason why these two steps are different is that the first one will require insertion of the corrected perturbation Anstze into the minimal ten-dimensional type IIB supergravity two-derivative part Eq.(19). The second step requires insertion of the uncorrected perturbation Anstze into the higher-curvature terms in ten dimensions.

Our plan here is to start from the corrected metric and solutions, then proposing an Anstze for the perturbations that may be inserted into Eq.(19).

As mentioned before, the only piece of the -action which affects the metric is the term. This induces the following corrected metric [35, 36, 37]

(28)

where we have used similar notation as for Eq.(7). The functions of in the above metric are

(29)

where there are the following exponents, which are functions of the radial coordinate

(30)

In addition, the radius of the black hole horizon gets corrections given by

(31)

has been already identified as the physical equilibrium temperature of the plasma. Thus, having obtained the corrected metric Eq.(28), we have to focus upon the appropriate perturbation Anstze. The vectorial perturbation we are interested in enters the metric and the solution, in contrast to the metric tensor perturbations - needed for mass-transport phenomena in the hydrodynamical regime of the plasma - the latter only enter the metric Ansatz, but not the Ansatz. This observation obviously makes the computation of the corrections to the mass-transport coefficients much more straightforward compared with the electric charge-transport coefficients as well as other plasma properties beyond the hydrodynamical domain.

We first obtain the kinetic term for the gauge fields. For this purpose we plug the corrected Ansatz into the two-derivative supergravity action Eq.(19). The metric Ansatz reads

(32)

where the metric of the unit five-sphere is given by

Notice that since we are only interested in the terms which are quadratic in the gauge-field perturbations we can write the Ansatz as follows

(33)

where is the Abelian field strength and is a deformation of the volume form of the metric of the -Schwarzschild black hole. We should mention that we are not interested in the part of which does not contain the vector perturbations because it only contributes to the potential of the metric, and is thus accounted for by the use of the corrected metric in the computation. The Hodge dual is taken with respect to the ten-dimensional metric, while denotes the Hodge dual with respect to the five-dimensional metric piece of the black hole. In addition, we have the usual definitions for the coordinates on the

(34)

By inserting these Ansätze into Eq.(19), and discarding all the higher (massive) Kaluza-Klein harmonics of the five-sphere, we get the following action for the zero-mode Abelian gauge field

(35)

Above we have written the Abelian field strength, defined as , the partial derivatives are , while , with and , are the Minkowski coordinates, and , which only involves the metric of -Schwarzschild black hole. Also notice that straightforwardly comes from the dimensional reduction [43]. The volume of the five-sphere has been included in .

Now, we should get the effect of the eight-derivative corrections of Eq.(25). In order to achieve this we must determine the five-dimensional operators that arise once the perturbed metric and five-form field strength Anstze are inserted into Eq.(25). As in [44], we use the uncorrected Anstze at this point. Indeed, we can do it because using the corrected ones generates terms of even higher order in . Clearly, the uncorrected Ansätze are derived from the ones displayed here by taking and . Next, we explain how to calculate the explicit contributions from the ten-dimensional operators, leading to the photoemission rates.

4.2 ’t Hooft coupling corrections to photoemission rates

In order to calculate the ’t Hooft coupling corrections to photoemission rates we now perform the explicit dimensional reduction on , including the leading type IIB string theory corrections discussed in the previous subsection. This is done along the lines of our previous work [2]333The main difference with respect to our previous calculation of the electrical conductivity of plasma in [2] is that while for the conductivity it is only needed to consider the dependence , for the photoemission rate it is necessary to consider the dependence which is not a trivial extension of our former calculations in [2]. Thus, having the dependence implies actually a much more complicated calculation.. For this purpose it is necessary to write explicitly all the terms of the full set of higher derivative ten-dimensional operators which come from the supersymmetric completion obtained in [25]. We use the definitions introduced in [27]

(36)

Thus, we can write the two contributions to the term as follows

(37)

Repeated indices means usual Lorentz contractions. In order to extract the quadratic terms in the vectorial fluctuations of the metric we should notice that they can straightforwardly be computed by expanding the ten-dimensional Weyl tensor as , where the sub-indices label the number of times that the Abelian gauge field occurs. Obviously, a similar expansion can be made for the tensor: . In addition, from a straightforward explicit calculation on the present background it can be shown that all the components of are zero. This fact is responsible of an important simplification of the actual computations. Also, is zero for any compactification which contains a five-dimensional Einstein manifold [28], and therefore it vanishes in the case we consider here.

Now, let us look at terms of the form ;

(38)

Their only possible contributions comes in fact from terms like , and .

Then, let us study operators like . We find a few contractions which can be collected in the following terms

The vanishing result of implies that terms like also vanish. Then, the only possible type of contribution from these terms is of the form . Making use of the same arguments all the terms like and include a factor and, therefore, are not present in a reduction upon a generic five-dimensional Einstein manifold [44].

Now, we proceed to explicitly calculate the operators above. Firstly, we must calculate the ten-dimensional Weyl tensor with and without vector fluctuations. Secondly, we need to obtain , and by its definition it can be separated into one piece which contains the covariant derivative, defined by

(40)

and a second piece which does not contain covariant derivatives which reads

(41)

So, we can write this tensor as .

Let us define . Thus, with the obvious meaning of the electric and magnetic contribution, for the electric part we have

(42)

where indicates the Hodge dual operation with respect to the -Schwarzschild black hole metric. It is convenient to split the electric part into the background plus a fluctuation,

(43)

and similarly for the magnetic terms. Therefore, in components we have

(44)

where is the determinant of the piece of the metric, in fact . The Hodge dual gives

(45)

Let us focus on the fluctuation. Actually, for this calculation we only need the gauge component , where in this notation and . Notice that if we were interested in the electrical conductivity it is enough to consider the dependence, which largely simplifies the calculation [2] in comparison with the actual calculation of the photoemission rates that we make in this work. Therefore, we have to deal with the following non-vanishing components of the two-form field strength: , and , all of them with the full dependence on and -coordinates. We use the following definition: .

So, the fluctuations of the electric part induce fluctuations in the Ramond-Ramond field strength which are given by

(46)
(47)
(48)

where

(49)
(50)
(51)

where the pairs are , , , and . The indices run over the coordinates of , and correspond to the coordinates and . The functions are:

(52)

The fluctuations on the magnetic part are obtained after performing the ten-dimensional Hodge dual operation on the corresponding electrical fluctuations above. We present the full expression in the appendix.

The kinetic term of the gauge field coming from the Ramond-Ramond five-form field strength becomes

(53)

which is exactly what is expected. Recall that the scalar curvature piece of the action gives , where denotes .

As we have seen in our previous paper [44], the eight-derivative corrections introduce a large number of higher-derivative operators after the compactification on a general five-dimensional Einstein manifold is done. We must take account of them properly to solve the equation of motion within perturbation theory. The situation is entirely analogous to that studied in [39], where the authors were concerned with the tensor perturbations of the metric, but the rationale is the same. We have discussed this for vectorial perturbations of the metric in [2, 44]. Lagrangian for the transverse mode reads

(54)