Massless bound-state excitations and the Schwinger mechanism in QCD

Massless bound-state excitations and the Schwinger mechanism in QCD


The gauge invariant generation of an effective gluon mass proceeds through the well-known Schwinger mechanism, whose key dynamical ingredient is the nonperturbative formation of longitudinally coupled massless bound-state excitations. These excitations introduce poles in the vertices of the theory, in such a way as to maintain the Slavnov-Taylor identities intact in the presence of massive gluon propagators. In the present work we first focus on the modifications induced to the nonperturbative three-gluon vertex by the inclusion of massless two-gluon bound-states into the kernels appearing in its skeleton-expansion. Certain general relations between the basic building blocks of these bound-states and the gluon mass are then obtained from the Slavnov-Taylor identities and the Schwinger-Dyson equation governing the gluon propagator. The homogeneous Bethe-Salpeter equation determining the wave-function of the aforementioned bound state is then derived, under certain simplifying assumptions. It is then shown, through a detailed analytical and numerical study, that this equation admits non-trivial solutions, indicating that the QCD dynamics support indeed the formation of such massless bound states. These solutions are subsequently used, in conjunction with the aforementioned relations, to determine the momentum-dependence of the dynamical gluon mass. Finally, further possibilities and open questions are briefly discussed.

12.38.Lg, 12.38.Aw, 12.38.Gc

I Introduction

The numerous large-volume lattice simulations carried out in recent years have firmly established that, in the Landau gauge, the gluon propagator and the ghost dressing function of pure Yang-Mills theories are infrared finite, both in  (1); (2); (3); (4); (5) and in  (6); (7); (8); (9). Perhaps the most physical way of explaining the observed finiteness of these quantities is the generation of a non-perturbative, momentum-dependent gluon mass (10); (11); (12); (13); (14); (15), which acts as a natural infrared cutoff. In this picture the fundamental Lagrangian of the Yang-Mills theory (or that of QCD) remains unaltered, and the generation of the gluon mass takes place dynamically, through the well-known Schwinger mechanism (16); (17); (18); (19); (20); (21); (22) without violating any of the underlying symmetries (for further studies and alternative approaches, see, e.g., (24); (25); (26); (27); (28); (23); (29)).

The way how the Schwinger mechanism generates a mass for the gauge boson (gluon) can be seen most directly at the level of its inverse propagator, , where is the dimensionless vacuum polarization. According to Schwinger’s fundamental observation, if develops a pole at zero momentum transfer (), then the vector meson acquires a mass, even if the gauge symmetry forbids a mass term at the level of the fundamental Lagrangian. Indeed, if , then (in Euclidean space) , and so the vector meson becomes massive, , even though it is massless in the absence of interactions (, (18); (19).

The key assumption when invoking the Schwinger mechanism in Yang-Mills theories, such as QCD, is that the required poles may be produced due to purely dynamical reasons; specifically, one assumes that, for sufficiently strong binding, the mass of the appropriate bound state may be reduced to zero (18); (19); (20); (21); (22). In addition to triggering the Schwinger mechanism, these massless composite excitations are crucial for preserving gauge invariance. Specifically, the presence of massless poles in the off-shell interaction vertices guarantees that the Ward identities (WIs) and Slavnov Taylor identities (STIs) of the theory maintain exactly the same form before and after mass generation (i.e. when the the massless propagators appearing in them are replaced by massive ones)  (10); (21); (22); (15). Thus, these excitations act like dynamical Nambu-Goldstone scalars, displaying, in fact, all their typical characteristics, such as masslessness, compositeness, and longitudinal coupling; note, however, that they differ from Nambu-Goldstone bosons as far as their origin is concerned, since they are not associated with the spontaneous breaking of any global symmetry (10). Finally, every such Goldstone-like scalar, “absorbed” by a gluon in order to acquire a mass, is expected to actually cancel out of the -matrix against other massless poles or due to current conservation (18); (19); (20); (21); (22).

The main purpose of the present article is to scrutinize the central assumption of the dynamical scenario outlined above, namely the possibility of actual formation of such massless excitations. The question we want to address is whether the non-perturbative Yang Mills dynamics are indeed compatible with the generation of such a special bound-state. In particular, as has already been explained in previous works, the entire mechanism of gluon mass generation hinges on the appearance of massless poles inside the nonperturbative three-gluon vertex, which enters in the Schwinger Dyson equation (SDE) governing the gluon propagator. These poles correspond to the propagator of the scalar massless excitation, and interact with a pair of gluons through a very characteristic proper vertex, which, of course, must be non vanishing, or else the entire construction collapses. The way to establish the existence of this latter vertex is through the study of the homogeneous Bethe-Salpeter equation (BSE) that it satisfies, and look for non-trivial solutions, subject to the numerous stringent constraints imposed by gauge invariance.

This particular methodology has been adopted in various early contributions on this subject; however, only asymptotic solutions to the corresponding equations have been considered. The detailed numerical study presented here demonstrates that, under certain simplifying assumptions for the structure of its kernel, the aforementioned integral equation has indeed non-trivial solutions, valid for the entire range of physical momenta. This result, although approximate and not fully conclusive, furnishes additional support in favor of the concrete mass generation mechanism described earlier.

The article is organized as follows. In Section II we set up the general theoretical framework related to the gauge-invariant generation of a gluon mass; in particular, we outline how the vertices of the theory must be modified, through the inclusion of longitudinally coupled massless poles, in order to maintain the WIs and STIs of the theory intact. In Section III we take a detailed look into the structure of the non-perturbative vertex that contains the required massless poles, and study its main dynamical building blocks, and in particular the transition amplitude between a gluon and a massless excitation and the proper vertex function (bound-state wave function), controlling the interaction of the massless excitation with two gluons. In addition, we derive an exact relation between these two quantities and the first derivative of the (momentum-dependent) gluon mass. Then, we derive a simple formula that, at zero momentum transfer, relates the aforementioned transition amplitude to the gluon mass. In the next two sections we turn to the central question of this work, namely the dynamical realization of the massless excitation within the Yang-Mills theory. Specifically, in Section IV we derive the BSE that the proper vertex function satisfies, and implement a number of simplifying assumptions. Then, in Section V we demonstrate through a detailed numerical study that the resulting homogeneous integral equation admits indeed non-trivial solutions, thus corroborating the existence of the required bound-state excitations. In Section VI we demonstrate with a specific example the general mechanism that leads to the decoupling of all massless poles from the physical (on-shell) amplitude. Finally, in Section VII we discuss our results and present our conclusions.

Ii General considerations

In this section, after establishing the necessary notation, we briefly review why the dynamical generation of a mass is inextricably connected to the presence of a special vertex, which exactly compensates for the appearance of massive instead of massless propagators in the corresponding WIs and STIs.

The full gluon propagator in the Landau gauge is defined as




is the usual transverse projector, and the scalar cofactor is related to the (all-order) gluon self-energy through


One may define the dimensionless vacuum polarization by setting so that (3) becomes


As explained in the Introduction, if develops at zero momentum transfer a pole with positive residue , then , and the gluon is endowed with an effective mass.

Alternatively, one may define the gluon dressing function as


In the presence of a dynamically generated mass, the natural form of is given by (Euclidean space)


where the first term corresponds to the “kinetic term”, or “wave function” contribution, whereas the second is the (positive-definite) momentum-dependent mass. If one insist on maintaining the form of (5) by explicitly factoring out a , then


and the presence of the pole, with residue given by , becomes manifest.

Figure 1: The “one-loop dressed” gluon contribution to the PT-BFM gluon self-energy. White (black) circles denote fully dressed propagators (vertices); a gray circle attached to the external legs indicates that they are background gluons. Within the PT-BFM framework these two diagrams constitute a transverse subset of the full gluon SDE.

Of course, in order to obtain the full dynamics, such as, for example, the momentum-dependence of the dynamical mass, one must turn eventually to the SDE that governs the corresponding gauge-boson self-energy (see Fig. 1). In what follows we will work within the specific framework provided by the synthesis of the pinch technique (PT)  (10); (30); (31); (32); (33); (34) with the background field method (BFM) (35). One of the main advantages of the “PT-BFM” formalism is that the crucial transversality property of the gluon self-energy , namely , is maintained at the level of the truncated SDEs (12); (36).

The Schwinger mechanism is integrated into the SDE of the gluon propagator through the form of the three-gluon vertex. In particular, as has been emphasized in some of the literature cited above (e.g.,(15)), a crucial condition for the realization of the gluon mass generation scenario is the existence of a special vertex, to be denoted by which must be completely longitudinally coupled, i.e. must satisfy


We will refer to this special vertex as the “pole vertex” or simply “the vertex ”.

The role of the vertex is indispensable for maintaining gauge invariance, given that the massless poles that it must contain in order to trigger the Schwinger mechanism, act, at the same time, as composite, longitudinally coupled Nambu-Goldstone bosons. Specifically, in order to preserve the gauge-invariance of the theory in the presence of masses, the vertex must be added to the conventional (fully-dressed) three-gluon vertex , giving rise to the new full vertex, , defined as


Gauge-invariance remains intact because satisfies the same STIs as before, but now replacing the gluon propagators appearing on their rhs by a massive ones; schematically, , where the former denotes the propagator given in (5), while the latter that of (6). In particular, in the PT-BFM framework that we employ, the vertex connects a background gluon () with two quantum gluons (), and is often referred to as the “BQQ” vertex. This vertex satisfies a (ghost-free) WI when contracted with the momentum of the background gluon, whereas it satisfies a STI when contracted with the momentum or of the quantum gluons. In particular,


where is the “ghost dressing function”, defined as , is the standard gluon-ghost kernel, and is the same as but with the external quantum gluon replaced by a background gluon. Similarly, is the dressing function of the self-energy connecting a background with a quantum gluon; is related to through the identity (37); (38)


The function is the scalar co-factor of the component of the special two-point function , defined as


Note finally that, in the Landau gauge, and are linked to by the exact (all-order) relation (39); (40); (41); (42)


to be employed in Subsection D.

Returning to the nonperturbative vertex , gauge invariance requires that it must satisfy the WI and STI of (10), with the replacement , e.g.,


exactly analogous expressions will hold for the STIs satisfied when contracting with the momenta or . Indeed, under this assumption, the full vertex will satisfy the same WI and STIs as the vertex before the introduction of any masses, but now with the replacement . Specifically, combining the first relation in (10) with (14), one obtains for the WI of ,


which is indeed the first identity in Eq. (10), with the aforementioned replacement enforced. The remaining two STIs are realized in exactly the same fashion.

It must be clear at this point that the longitudinal nature of , combined with the WI and STIs that it must satisfy, lead inevitably to the appearance of a massless pole, as required by the Schwinger mechanism. For example, focusing only on the -channel, the simplest toy Ansatz for the vertex is


which has a pole in and satisfies (14). Of course, poles associated to the other channels ( and ) will also appear, given that must also satisfy the corresponding STIs with respect to and .

Iii The pole vertex: structure and properties

In this section we have a detailed look at the structure of the special vertex . In particular, we identify the diagrammatic origin and field-theoretic nature of the various quantities contributing to it, and specify the way it enters into the SDE of the full vertex , defined in Eq. (9). In addition, we will derive an exact relation between the most important component of this vertex and the derivative of the momentum-dependent gluon mass.

iii.1 General structure of the vertex

The main characteristic of the vertex , which sharply differentiates it from ordinary vertex contributions, is that it contains massless poles, originating from the contributions of bound-state excitations. Specifically, all terms of the vertex are proportional to , , , and products thereof. Such dynamically generated poles are to be clearly distinguished from poles related to ordinary massless propagators, associated with elementary fields in the original Lagrangian.

Figure 2: The SDE for the BQQ vertex which connects a background gluon () with two quantum gluons ().

In general, when setting up the usual SDE for any vertex (see, for example, Fig. 2), a particular field (leg) is singled out, and is connected to the various multiparticle kernels through all elementary vertices of the theory involving this field (leg). The remaining legs enter into the various diagrams through the aforementioned multiparticle kernels (black circles in graphs in Fig. 2), or, in terms of the standard skeleton expansion, through fully-dressed vertices (instead of tree-level ones). For the case of the vertex that we consider here [shown in Fig. 2], it is convenient (but not obligatory) to identify as the special leg the background gluon, carrying momentum . Now, with the Schwinger mechanism turned off, the various multiparticle kernels appearing in the SDE for the vertex have a complicated skeleton expansion (not shown here), but their common characteristic is that they are one-particle-irreducible with respect to cuts in the direction of the momentum ; thus, a diagram such as the of Fig. 3 is explicitly excluded from the (gray) four-gluon kernel, and the same is true for all other kernels.

Figure 3: (A) A diagram that does not belong to the standard kernel. (B) The gray kernel (regular part with respect to , and the composite massless excitation in the -channel. (C) The part of the vertex.

When the Schwinger mechanism is turned on, the structure of the kernels is modified by the presence of composite massless excitation, described by a propagator of the type , as shown in Fig. 3. The sum of such dynamical terms, coming from all multiparticle kernels, shown in Fig. 4 constitutes a characteristic part of the vertex , to be denoted by in Eq. (20), namely the part that contains at least a massless propagator . The remaining parts of the vertex , to be denoted by in Eq. (21), contain massless excitations in the other two channels, namely and (but no ), and originate from graphs such as () of Fig. 3. Indeed, note that the kernel () is composed by an infinite number of diagrams, such as (), containing the full vertex ; these graphs, in turn, will furnish terms proportional to and [e.g., graph ()].

In order to study further the structure and properties of the vertex , let us first define the full vertex , given by


with satisfying Eq. (8). Using a general Lorentz basis, we have the following expansion for in terms of scalar form factors,


According to the arguments presented above, may be decomposed into






All form-factors of (namely ) must contain a pole , while some of them may contain, in addition, and poles. On the other hand, none of the form-factors of (namely ) contains poles, but only and poles.

In what follows we will focus on , which contains the explicit -channel massless excitation, since this is the relevant channel in the SDE of the gluon propagator, where will be eventually inserted [graph in Fig. 1]. In fact, with the two internal gluons of diagram () in the Landau gauge, we have that


so that the only relevant form factors are and .

Figure 4: (A) The vertex is composed of three main ingredients: the transition amplitude, , which mixes the gluon with a massless excitation, the propagator of the massless excitation, and the (massless excitation)–(gluon)–(gluon) vertex. (B) The Feynman rules (with color factors included) for (i) the propagator of the massless excitation and (ii) the “proper vertex function”, or, “bound-state wave function”, . (C) The various appearing in Eq. (30).

At this point we can make the nonperturbative pole manifest, and cast in the form of Fig. 4, by setting


where the nonperturbative quantity


is the effective vertex (or “proper vertex function” (19)) describing the interaction between the massless excitation and two gluons. In the standard language used in bound-state physics, represents the “bound-state wave function” (or “BS wave function”) of the two-gluon bound-state shown in of Fig. 3; as we will see in Section IV, satisfies a (homogeneous) BSE. In addition, is the propagator of the scalar massless excitation. Finally, is the (nonperturbative) transition amplitude introduced in Fig. 4, allowing the mixing between a gluon and the massless excitation. Note that this latter function is universal, in the sense that it enters not only in the pole part associated with the three-gluon vertex, but rather in all possible such pole parts associated with all other vertices, such as the four-gluon vertex, the gluon-ghost-ghost vertex, etc (see panel C in Fig. (4)).

Evidently, by Lorentz invariance,


and the scalar cofactor, to be referred to as the “transition function”, is simply given by


so that


Note that, due to Bose symmetry (already at the level of ) with respect to the interchange and , we must have


which implies that


Finally, in principle, all other elementary vertices of the theory may also develop pole parts, which will play a role completely analogous to that of in maintaining the corresponding STIs in the presence of a gluon mass. Specifically, in the absence of quarks, the remaining vertices are the gluon-ghost-ghost vertex, , the four-gluon vertex , and the gluon-gluon-ghost-ghost vertex , which is particular to the PT-BFM formulation. The parts of their pole vertices containing the , denoted by , , and , respectively, will all assume the common form


where the various are shown in panel C of Fig. 4.

iii.2 An exact relation

The WI of Eq (14) furnishes an exact relation between the dynamical gluon mass, the transition amplitude at zero momentum transfer, and the form factor . Specifically, contracting both sides of the WI with two transverse projectors, one obtains,


On the other hand, contracting the full expansion of the vertex (18) by these transverse projectors and then contracting the result with the momentum of the background leg, we get


where the relation of Eq (27) has been used. Thus, equating both results, one arrives at


The above relations, together with those of Eq. (27), determine exactly the form factors and of the vertex , namely


We will now carry out the Taylor expansion of both sides of Eq (33) in the limit . To that end, let consider the Taylor expansion of a function around (and ). In general we have


where the prime denotes differentiation with respect to and subsequently taking the limit , i.e.


Now, if the function is antisymmetric under , as happens with the form factors , then ; thus, for the case of the form factors in question, the Taylor expansion is ()


Using Eq (37), and the corresponding expansion for the rhs,


assuming that the is finite, and equating the coefficients in front of , we arrive at (Minkowski space)


We emphasize that this is an exact relation, whose derivation relies only on the WI and Bose-symmetry that satisfies, as captured by Eq. (14) and Eq. (29), respectively. The Euclidean version of Eq. (39) is given in Eq. (73).

iii.3 “One-loop dressed” approximation for the transition function

We will next approximate the transition amplitude , connecting the gluon with the massless excitation, by considering only diagram in Fig. 4, corresponding to the gluonic “one-loop dressed” approximation; we will denote the resulting expression by .

In the Landau gauge, is given by


where the origin of the factor is combinatoric, and is the standard three-gluon vertex at tree-level,


To determine the corresponding transition function from Eq. (26), use that


to write


In the last step we have used the property of Eq. (28) in order to interchange the arguments of , so that the Taylor expansion of Eq. (37) may be applied directly; this accounts for the additional minus sign. Then, after the shift , and further use of Eq. (28), becomes


To obtain the limit of as , we will employ Eq. (37) for and , as well as


Observe that only the zeroth order term of , namely , contributes in this expansion. Then, using spherical coordinates to write , and the integral


the in Eq. (44) becomes in the limit (in )


Then, partial integration yields


and finally one arrives at (Minkowski space)


The Euclidean version of this equation, Eq. (75), will be used in Section V.

We end this subsection with a comment on the dimensionality of the various form factors. The vertex has dimension , and so , and are dimensionless, while the remaining form factors have dimension . The integral has the same dimension as , and as a result, in order to keep dimensionless, must have dimensions of .

iii.4 Relating the gluon mass to the transition function

In this subsection we show how the vertex gives rise to a gluon mass when inserted into the corresponding SDE. We will restrict ourselves to the two diagrams shown in Fig. 1, and will finally express exclusively in terms of , which, in turn, depends on the existence of through Eq. (49).

In the PT-BFM scheme, the SDE of the gluon propagator in the Landau gauge assumes the form


The most straightforward way to relate the gluon mass to the transition function is to identify, on both sides of (50), the co-factors of the tensorial structure which survive the limit , and then set them equal to each other. Making the usual identification (in Minkowski space) , it is clear that lhs of (50) furnishes simply


It is relatively straightforward to recognize that the analogous contribution from the rhs comes from the standard “squared” diagram, shown in Fig. 5. Specifically, the starting expression is


where, as explained earlier, the (all order) vertex has been replaced by , and the ellipses denote terms that, in the kinematic limit considered, do not contribute to the specific structure of interest.

Figure 5: The “squared” diagram.

The relevant contribution originates from the part containing the vertex , to be denoted by ; it is represented by the diagram in Fig. 5. In particular, by virtue of Eq. (22), we have


where in the second line we have used Eq. (23) [ with ], and Eq. (40) in the third line.

Thus, using the fact that, since  (42), from the identity of Eq. (13) we have that , then the rhs of (50) becomes


We next go to Euclidean space, following the usual rules, and noticing that, due to the change we have ; so, equating (51) and (54) we obtain (suppressing the index “E”)


Note that the so obtained is positive-definite. We emphasize that the relation of Eq. (55) constitutes the (gluonic) “one-loop dressed” approximation of the complete relation; indeed, both the SDE used as starting point as well as the expression for are precisely the corresponding “one-loop dressed” contributions, containing gluons (but not ghosts).

Finally, let us consider the exact relation (15)


expressing the dynamical mass of the standard gluon propagator in terms of the corresponding mass, , of the PT-BFM gluon propagator [usually denoted by ] in the same gauge [in this case, in the Landau gauge]. At this relation reduces to , so that Eq. (55) may be alternatively written as


Interestingly enough, when written in this form, the mass formula derived from our SDE analysis coincides with the one obtained for the photon mass in the Abelian model of Jackiw and Johnson (Eq. (2.12) in (18)). In addition, this last form facilitates the demonstration of the decoupling of the massless excitation from the on-shell four-gluon amplitude (see Section VI).

In principle, the analysis presented above may be extended to include the rest of the graphs contributing to the gluon SDE, invoking the corresponding pole parts of the remaining vertices; however, this lies beyond the scope of the present work.

Iv BS equation for the bound-state wave function

As has become clear in the previous section, the gauge boson (gluon) mass is inextricably connected to the existence of the quantity . Indeed, if were to vanish, then, by virtue of (49) so would , and therefore, through (55) we would obtain a vanishing . Thus, the existence of is of paramount importance for the mass generation mechanism envisaged here; essentially, the question boils down to whether or not the dynamical formation of a massless bound-state excitation of the type postulated above is possible. As is well-known, in order to establish the existence of such a bound state one must (i) derive the appropriate BSE for the corresponding bound-state wave function, , (or, in this case, its derivative), and (ii) find non-trivial solutions for this integral equation.

To be sure, this dynamical equation will be derived under certain simplifying assumptions, which will be further refined in order to obtain numerical solutions. We emphasize, therefore, that the analysis presented here is meant to provide preliminary quantitative evidence for the realization of the dynamical scenario considered, but cannot be considered as a conclusive demonstration.

Figure 6: The complete BSE for the full three gluon vertex .

The starting point is the BSE for the vertex , shown in Fig. 6. Note that, unlike the corresponding SDE of Fig. 2, the vertices where the background gluon is entering (carrying momentum ) are now fully dressed. As a consequence, the corresponding multiparticle kernels appearing in Fig. 6 are different from those of the SDE, as shown in Fig. 7.

Figure 7: (A) Schematic relation between the SDE and BSE kernels. (B) Example of a diagram not contained in the corresponding BSE kernel, in order to avoid over counting.
Figure 8: (A) The separation of the vertex in regular and pole parts. (B) The BSE for the bound-state wave function .

The general methodology of how to isolate from the BSE shown in Fig. 6 the corresponding dynamical equation for the quantity has been explained in (19); (22). Specifically, one separates on both sides of the BSE equation each vertex (black circle) into two parts, a “regular” part and another containing a pole ; this separation is shown schematically in Fig. 8. Then, the BSE for is obtained simply by equating the pole parts on both sides. Of course, for the case we consider the full implementation of this general procedure would lead to a very complicated structure, because, in principle, all fully dressed vertices appearing on the rhs of Fig. 6 may contain pole parts [i.e., not just the three-gluon vertex of (a) but also those in (b), (c), and (d)]. Thus, one would be led to an equation, whose lhs would consist of , but whose rhs would contain the together with all other similar vertices, denoted by in Eq. (30). Therefore, this equation must be supplemented by a set of analogous equations, obtained from the BSEs of all other vertices appearing on the rhs of Fig. 6 [i.e., those in (b), (c), (d) ]. So, if all vertices involved contain a pole part, one would arrive at a system of several coupled integral equations, containing complicated combinations of the numerous form factors composing these vertices (see, for example, Fig. 11 in (22)).

It is clear that for practical purposes the above procedure must be simplified to something more manageable. To that end, we will only consider graph (a) on the rhs of Fig. 6, thus reducing the problem to the treatment of a single integral equation.

Specifically, the BSE for is given by [see Fig. 6]

Figure 9: The Feynman diagrams considered for the BS kernel. The interaction vertices are approximated by their tree level values, while the internal gluon propagators are fully dressed.

In addition, we will approximate the four-gluon BS kernel by the lowest-order set of diagrams shown in Fig. 9, where the vertices are bare, while the internal gluon propagators are fully dressed.

To proceed further, observe that the diagram does not contribute to the BSE, because the color structure of the tree-level four-gluon vertex vanishes when contracted with the color factor of the . Diagrams and are equal, and are multiplied by a Bose symmetry factor of . So, in this approximation, the BS kernel is given by


where is the tree-level value of the three gluon vertex. So, using this kernel and setting the gluon propagators in the Landau gauge, the BSE becomes


where we have cancelled out a color factor from both sides.

Let us focus on the lhs of Eq. (60). Using the Taylor expansion in Eq. (37), the fact that [see Eq. (33)], and multiplying by a transverse projector we obtain,


Next, let us denote by the rhs of Eq. (60). Inserting the bare value for the three gluon vertices, multiplying by the transverse projector, and using the Taylor expansions in Eq. (37) and (45), after standard manipulations one obtains the result


where we have defined the kernel