Quantum field theoretic properties of Lorentz-violating operators of nonrenormalizable dimension in the photon sector
In the context of the nonminimal Standard-Model Extension a special subset of the CPT-even higher-dimensional operators in the photon sector is discussed from a quantum-field theoretical point of view. The modified dispersion laws, photon polarization vectors plus the gauge field propagator are obtained and their properties are analyzed. It is demonstrated that for certain sectors of the modified theory a puzzle arises for the optical theorem at tree-level. This is followed by a discussion of how it can be interpreted and resolved at first order Lorentz violation. Furthermore the commutator of two gauge fields that are evaluated at different spacetime points is obtained and discussed. The structure of the theory is shown to resemble the structure of the modification based on the corresponding dimension-4 operator. However some properties are altered due to the nonrenormalizable nature of the theory considered. The results provide more insight into the characteristics of Lorentz-violating quantum field theories that rest upon contributions of nonrenormalizable dimension.
pacs:11.30.Cp, 14.70.Bh, 03.70.+k, 11.15.Bt
Over the past 15 years the study of Lorentz invariance violation both in theory and experiment has become an important field. The foundations were laid by the seminal papers Kostelecky:1988zi ; Kostelecky:1991ak ; Kostelecky:1994rn in which it was shown that a violation of Lorentz symmetry can emerge in certain scenarios of string theory. In addition, Lorentz violation may arise in many other interesting contexts such as a nontrivial structure of spacetime (spacetime foam) Wheeler:1957mu ; Hawking:1979zw ; Bernadotte:2006ya ; Klinkhamer:2003ec , noncommutative field theories Carroll:2001ws , loop quantum gravity Gambini:1998it ; Bojowald:2004bb , and quantum field theories on spacetimes with a nontrivial topological structure Klinkhamer:1999zh ; Klinkhamer:1998fa .
In principle it is assumed that a low-energy effective description of quantum gravity phenomena can be considered as an expansion in energy over a mass scale, which is probably related to the Planck scale. The leading-order term in such an expansion comprises the ordinary Standard Model of elementary particle physics plus General Relativity. The next-to-leading order term is the minimal Standard-Model Extension (SME) Colladay:1998fq , which is a framework to study and test Lorentz violation at energies much smaller than the Planck scale. The minimal SME includes all Lorentz-violating operators that are invariant under the gauge group of the Standard Model plus power-counting renormalizable. Since gravity itself is nonrenormalizable one may expect higher-order terms in the expansion to be made up of operators of nonrenormalizable dimension. These are included in the nonminimal SME.
A special sector of the nonrenormalizable SME forms the basis of the current paper. A necessary (but not sufficient) criterion for a renormalizable, interacting quantum field theory is that it only contains products of field operators that have a mass dimension of four or less. Operators with mass dimension of at least five are called higher-dimensional. In the early days of the development and understanding of renormalization many theorists considered such quantum field theories with antipathy. However, this point of view has changed. Nowadays nonrenormalizable quantum field theories are powerful tools in both high-energy and condensed matter physics. Such theories have significance as long as they are considered as an effective theory only valid within a certain energy range Georgi:1994qn ; Pich:1998xt .
There are many examples for effective theories: Fermi’s theory of the weak interaction Fermi:1934sk , Euler-Heisenberg theory Heisenberg:1935qt , chiral perturbation theory, heavy quark effective theory, etc. (for the latter two cf. the review Pich:1998xt ). Even the Standard Model of elementary particle physics can be considered as an effective (though renormalizable) theory and practically all condensed matter theories are effective ones. To illustrate why these are successful we consider Euler-Heisenberg theory as an example for an effective theory in the photon sector.
Let us assume that we only knew about classical physics, e.g., the Lagrange density of classical electrodynamics, which is proportional to a bilinear combination of electromagnetic field strength tensors, , and leads to Maxwell’s equations. Based on Lorentz invariance, nothing would forbid us to add higher-dimensional terms to this Lagrange density. Two of the possible terms are and with the dual field strength tensor , where each of them is multiplied with an unknown coefficient of mass dimension . These would then lead to nonlinear versions of Maxwell’s equations describing light-by-light scattering. The latter is certainly not a classical but a quantum theoretical phenomenon. Hence even without any knowledge of quantum theory we could gain an understanding of it if we were able to determine the unknown coefficients by experiment. However, since we know about quantum theory, these coefficients can be calculated in perturbative quantum electrodynamics (QED) and they are inversely proportional to the fourth power of the electron mass. Although Euler-Heisenberg theory is nonrenormalizable, it gives a good description for quantum effects in electrodynamics as long as the the photon energy is much smaller than the electron mass. The nonrenormalizable nature of the theory is revealed since the cut-off dependence of scattering quantities cannot be removed by a redefinition of the theory parameters. When the cut-off reaches the electron mass the higher-dimensional terms can be as large as the renormalizable ones and the validity of the theory breaks down.
The example given shall demonstrate how useful nonrenormalizable field theories can still be. This was certainly one motivation for Kostelecký and Mewes to include higher-dimensional operators in the photon Kostelecky:2009zp , neutrino Kostelecky:2011gq , and the fermion sector Kostelecky:2013rta of the minimal SME, which leads us to the nonminimal SME. Since Lorentz violation is supposed to originate from physics at the Planck scale, these terms of nonrenormalizable dimension are probably suppressed by the Planck mass. The theory is applicable as long as particle momenta do not lie in the order of magnitude of this scale.
The current paper shall provide a better insight on the quantum field theoretical properties of such Lorentz-violating nonrenormalizable theories. Here the focus is on the nonminimal, CPT-even photon sector whose terms are classified in Kostelecky:2009zp . In Sec. II the action of the theory considered will be introduced and its general properties will be discussed. The framework is then restricted to a particular subset of Lorentz-violating parameters. Based upon this modification, the modified dispersion relations of electromagnetic waves are obtained and investigated in Sec. III, which is followed by the calculation of the polarization vectors and the photon propagator in Sec. IV. Using these results the optical theorem at tree-level will be checked in Sec. V. It is demonstrated that a puzzle arises for certain sectors of the theory and how to resolve it at leading order Lorentz violation. Section VI is dedicated to studying the properties of the gauge potential commutator of the theory with the goal to get some understanding of its causal structure. The results are concluded in Sec. VII. Calculational details are relegated to Apps. A and B. Natural units with will be used throughout the paper unless stated otherwise.
Ii Cpt-even photon sector of the nonminimal Standard-Model Extension
Within this paper modified Maxwell theory shall be considered, which is the CPT-even modification of the SME photon sector. Including the dimension-4 and all higher-dimensional contributions this theory is defined by the following action:
The action is written in terms of the electromagnetic field strength tensor of the U(1) gauge field . The fields are defined on Minkowski spacetime with coordinates and metric . The Lagrange density of Eq. (2.1b) is decomposed into the standard Maxwell term and the Lorentz-violating modification. The latter involves the background coefficients , which transform covariantly with respect to observer Lorentz transformations and are fixed with respect to particle Lorentz transformations.
Besides the dimension-4 modified Maxwell term (cf. Refs. ChadhaNielsen1983 ; Colladay:1998fq ; KosteleckyMewes2002 ) the background coefficients involve all higher-dimensional operators with even operator dimension . The terms of nonrenormalizable dimension are characterized by additional derivatives. Each increase of the operator dimension by two involves two additional derivatives that are contracted with an appropriate Lorentz-violating coefficient with two additional indices. The mass dimension of this coefficient is decreased by two to compensate the mass dimensions of the derivatives. Transforming these coefficients to momentum space with the four-momentum yields:
Hence the scales of the individual contributions in the expansion of can be made more transparent with the following symbolic notation:
where denotes the order of magnitude of the Lorentz-violating coefficients and are the coefficients associated with the dimension-4 operator, . The quantities (for even ) have mass dimension and the variable denotes a momentum scale. A necessary condition for this expansion to be well-defined is that . The leading dimension-4 operator is often called “marginal” and the subleading terms are named “irrelevant.”
Equation (2.3) means that the marginal operator is dominant as long as . However, the larger the momentum of a photon the more important are the higher-dimensional operators, which is why the expression “irrelevant” can be misleading in this case. Keep in mind that the validity of the effective theory breaks down when approaches an order of magnitude such that . Then all terms in the expansion above become equally important and it is not supposed to converge any more.
The properties of the quantum field theory based on the marginal operator and all the irrelevant ones set to zero have been investigated in the series of papers Casana-etal2009 ; Casana-etal2010 ; Klinkhamer:2010zs ; Schreck:2011ai ; Schreck:2013gma . The current goal is to understand the implications of including some of the higher-dimensional ones. To keep the calculations feasible we restrict the analysis to a particular subset of operators. It is natural to consider the first of the higher-dimensional operators that has mass dimension six where all remaining ones are set to zero:
There is a generalization of the nonbirefringent ansatz of the dimension-4 operator BaileyKostelecky2004 ; Altschul:2006zz . In particular for the dimension-6 operator it is given by Kostelecky:2009zp
with the Minkowski metric and a four-tensor . Now we want to restrict ourselves to the case that is equivalent to the isotropic sector of the dimension-4 operator. The corresponding is a -matrix that has the same form as the respective matrix Kaufhold:2007qd ; Casana-etal2010 of the isotropic sector mentioned. The isotropic dimensionless coefficient of the dimension-4 operator is replaced by the combination that now appears in each entry of the matrix:
with the timelike four-vector . All remaining Lorentz-violating coefficients are assumed to vanish. The coefficients have mass dimension . The minus sign in their index was added to distinguish them from the other set of isotropic parameters, . In the context of the coefficients , which are contained in the dimension-4 operator, corresponds to the double trace . Since the latter can be removed by a field redefinition Colladay:1998fq , does not describe any physics when the theory is restricted to the marginal operator. On the contrary, it can lead to physical effects for the higher-dimensional operators Kostelecky:2009zp , but it is discarded here for simplicity.
In what follows, the Lorentz-violating nonminimal SME sector characterized by Eqs. (2.1), (2.4), (II), and (2.6) shall be studied. Instead of only the coefficient for the marginal operator there now exists a -matrix that makes up this sector. Since this matrix is combined with the symmetric two-tensor , its antisymmetric part can be discarded. Therefore, is assumed to be completely symmetric. Furthermore, antisymmetrization on any triple of indices of produces zero Kostelecky:2009zp , which reduces the number of independent coefficients further. However these additional restrictions are not important for the current paper, and they are not considered.
Certain properties of the modified electrodynamics cannot be investigated without a coupling to matter. Therefore, the modified free theory is minimally coupled to a standard Dirac theory of spin-1/2 fermions with charge and mass . This results in a Lorentz-violating extended QED that is defined by the following action:
The latter action contains the standard Dirac matrices with the Clifford algebra and it is written such that the respective Lagrange density is Hermitian.
Iii Dispersion relations
The first step is to obtain the modified dispersion relations of electromagnetic waves. The field equations for the theory based on the higher-dimensional operators have the same form as those of the dimension-4 CPT-even extension Colladay:1998fq ; KosteleckyMewes2002 ; BaileyKostelecky2004 . They are given by:
with the four-momentum . The modified dispersion relations are the conditions that have to be fulfilled by the four-momentum such that Eq. (3.1) has nontrivial solutions for . They follow from with the matrix given in Eq. (3.1). To obtain the dispersion laws it is convenient to divide the matrix into three parts:
The first contains the single coefficient that appears together with two time derivatives. For this reason it will be denoted as the “temporal part.” The second sector is characterized by the three mixed coefficients for , 2, 3 that are combined with one time and one spatial derivative. Hence we call it the “mixed part.” Finally, the third sector is made up of the six spatial coefficients for , , 2, 3, whereby it is named the “spatial part.” The following investigations will be performed for these three sectors separately.
iii.1 Temporal part
In this case all coefficients are set to zero expect of . The determinant of involves a biquadratic polynomial whose solutions with respect to correspond to the two physical dispersion relations that are given as follows:
Note that has mass dimension whereby it always occurs in combination with to produce a dimensionless quantity. Since the second square root appears with two different signs there are two distinct dispersion relations with different phase velocities. To get a better insight in this issue the following expansions for both dispersion laws are given for :
The first dispersion law is a perturbation of the standard dispersion relation , whereas the second does not have an existing limit for . Such a behavior does not occur for the sectors of the dimension-4 operators that were considered in Casana-etal2009 ; Casana-etal2010 ; Klinkhamer:2010zs ; Schreck:2011ai ; Schreck:2013gma . The existence of traces back to treatment of the Lorentz-violating extension as an effective field theory. According to Kostelecky:2009zp such dispersion laws are neglected since they do not arise as a small perturbation from the standard relations. They must be considered as Planck scale effects. If is indeed nonzero in nature, modes that are associated with may become especially important if the momentum approaches the Planck scale. However keep in mind that we are dealing with an effective field theory whose applicability is expected to break down for momenta in the vicinity of the Planck scale (see also the discussion at the end of Sec. (IIc) in Kostelecky:2009zp ).
For the reasons mentioned, modified dispersion relations that are a perturbation of the standard one will be called “perturbed” and the others, which are not a perturbation, will be referred to as “spurious.” Note that the spurious dispersion relation of Eq. (3.4b) is, in fact, associated with one of the transverse, propagating modes. This can be shown with the modified Coulomb and Ampère law according to Colladay:1998fq . Using the latter procedure unphysical dispersion laws being associated to the scalar and longitudinal mode can be identified and discarded. This procedure cannot be applied to remove the spurious dispersion law, though.
iii.2 Mixed part
For the mixed part the determinant of is more complicated and contains a third order polynomial in . Note that this is the first of all sectors of modified Maxwell theory studied so far where the physical dispersion laws result from a polynomial of this degree. This renders the calculation of the dispersion relations more complicated in comparison to the aforementioned sectors. With the transformation
and an additional multiplication with the third order polynomial can be recast in the form with
where a summation over , 2, 3 is understood and will be needed below. Two of the three zeros of this polynomial with respect to that are transformed back to via Eq. (3.5) correspond to the dispersion laws of the transverse degrees of freedom. They are given by:
with and defined by Eq. (3.6a). Dependent on the sign of the functions , , , and one of these solutions is a perturbation of the standard dispersion law and the other one is a spurious dispersion relation similar to Eq. (3.4b). For and (where is negative for this choice) or , , and the only dispersion relation being a perturbation of the standard dispersion law is , which can then be rewritten as follows:
For , , and or , (where is negative in this case) or , , and only is such a perturbation, which is rearranged to give:
Both and have been recast, from which it can be shown that they both are real quantities. The cubic roots in Eq. (3.8a) give opposite imaginary parts that cancel in the sum. The second dispersion law of Eq. (3.9a) is manifestly real for . For the real parts arising in each of the two terms in the square brackets cancel, which gives a purely imaginary result. Combining it with the imaginary result from the square root outside of the brackets leads to a real quantity. These properties are not directly evident from Eqs. (3.7a), (3.7b).
For , , and the polynomial does not have a real and positive zero. Hence this choice does not lead to a modified dispersion law. Finally, Tab. 1 shows the regions of that result in nonnegative , , and .
iii.3 Spatial part
Finally, the case is considered where all coefficients are assumed to vanish if they contain at least one Lorentz index that is equal to zero. The physical dispersion law then results from a polynomial of second degree and can be cast in the following form:
Note the similarity to the isotropic dispersion relation when considering a nonvanishing dimension-4 operator with the isotropic coefficient :
Contrary to the previous cases, there only exists a single dispersion law, which is a Lorentz-violating perturbation of the standard dispersion relation. Hence for the spatial part of the dimension-6 operator the modified dispersion law has the same form as for the isotropic dimension-4 operator with replaced by . The physical dispersion relations found for the spatial and the mixed sector are no longer isotropic. This shows that higher-dimensional operators of the isotropic CPT-even modification of the photon sector can deliver anisotropic contributions to the dispersion relations.
A last comment concerns the degeneracy of the transverse dispersion relations for all three sectors previously considered. Both the perturbed and the spurious dispersion laws have a twofold degeneracy, i.e., they appear as a double zero of the determinant of the matrix in Eq. (3.1). The latter degeneracy reflects the degeneracy of the quantum-mechanical photon state. It is important that the degeneracy of the perturbed dispersion law is still twofold despite the occurrence of the spurious dispersion relations. The reason is that the photon state degeneracy goes in many physical quantities, e.g., Planck’s radiation law. Hence if it was modified, Planck’s law would change as well and the limit of vanishing Lorentz-violating coefficients would not describe the experimental measurements correctly.
Iv Polarization vectors and the propagator
The CPT-even Lorentz-violating modification considered is based on a higher-dimensional operator. Due to the additional derivatives that are combined with this operator it is interesting to examine the quantum-field theoretic properties of the modification. To do so, the modified polarization vectors and the propagator are needed and they are obtained as follows.
The propagator of a quantized field is an important object for studying the properties of the underlying quantum field theory. It is the Green’s function of the free-field equations of motion, i.e., the inverse of the differential operator that appears in these equations. However, due to the infinite number of gauge degrees of freedom of the photon field an inverse does not exist as long as no gauge fixing condition is imposed. For all cases of the CPT-even dimension-4 operator considered so far Casana-etal2009 ; Casana-etal2010 ; Klinkhamer:2010zs ; Schreck:2011ai ; Schreck:2013gma , Feynman gauge Veltman1994 ; ItzyksonZuber1980 ; PeskinSchroeder1995 has proven to be a convenient gauge choice. Hence, this gauge choice will be implemented here as well. In practice this is done by adding the following gauge-fixing term to the Lagrange density:
By partial integration the action of the modified photon sector can be written as follows:
|with the differential operator|
Transforming to momentum space leads to
Now the system of equations must be solved where is the propagator in momentum space. To understand the structure of the propagator it must be expressed in a covariant form using the four-vectors and two-tensors that are available in this context. This is the metric tensor , the four-vector , and the preferred spacetime direction , which appears in Eq. (2.6). The isotropic case based on the dimension-4 operator is characterized by . It is assumed that is the only preferred direction that plays a role for the dimension-6 operator as well. For this reason the following ansatz is made for the propagator:
The propagator coefficients , , plus the scalar part depend on the four-momentum components: , etc. Since the propagator is a symmetric two-tensor, the ansatz has to respect this property. This can be checked to be the case in Eq. (4.15). Due to this symmetry, from the original 16 equations only ten have to be solved to obtain the propagator coefficients plus the scalar part.
The propagator of a quantum field describes its off-shell properties. To understand its on-shell characteristics the dispersion relations are needed plus — in case of the photon field — the corresponding polarization vectors. The modified dispersion laws were already obtained in Sec. III. The polarization vectors will be determined as follows. These form a set of four four-vectors that is a basis of Minkowski spacetime. Only two of them describe physical, i.e., transverse photon polarization states where the remaining two correspond to scalar and longitudinal degrees of freedom. The transverse photon polarization vectors are solutions of the field equations (3.1) with to be replaced by the physical dispersion laws. For the temporal, the mixed, and the spatial sector they can be chosen as follows where is a general three-momentum:
with a normalization . The latter is an additional normalization, which is not related to the requirement that the scalar product of a polarization vector with itself is equal to one. On the contrary, it has to be determined from the 00-component of the energy-momentum tensor (given by Eq. (36) in Colladay:1998fq ) whose expectation value must correspond to the modified physical photon dispersion law. The procedure is described in App. A in detail. Note that besides the appearance of , the polarization vectors are completely standard. It can be checked that they are orthogonal to each other and each is orthogonal to the momentum three-vector. So they are interpreted as the physical transverse polarizations.
Equation (4.16) provides the polarization vectors of both the perturbed and the spurious dispersion relation. The reason is that the degeneracy of each dispersion law is still twofold as in the standard theory (cf. the discussion at the end of Sec. III).
As a next step the polarization sum
is computed where the bar means complex conjugation. To investigate the properties of the theory it is reasonable to write in a covariant form similar to the ansatz of Eq. (4.15), which was made for the propagator:
Here , , , and are unknown coefficients to be determined by comparing the ansatz to the explicit expression of Eq. (4.17) that is constructed with the polarization vectors of Eq. (4.16). Since the polarization sum is symmetric such as the propagator this leads to ten equations that must be fulfilled.
iv.1 General results
The propagator coefficients plus its scalar part can be computed for all ten Lorentz-violating coefficients at once. Introducing the short-hand notation one obtains:
Note that this propagator has the same structure as the propagator that is based on the isotropic CPT-even dimension-4 operator with the replacement . The minus sign in the definition of above originates from the minus sign that emerges when transforming the two additional derivatives of the dimension-6 operator to momentum space (cf. Eq. (2.2) for the general case of the dimension- operator and Eq. (2.4) for the dimension-6 operator). I anticipate that the propagator for the full dimensional expansion has exactly this form with to be replaced by
The coefficients of the polarization sum of Eq. (4.18) can be stated as
where has to be replaced by the respective physical dispersion law. Furthermore, the normalization of the polarization vectors is given by the following general expression:
Three remarks are in order. First, the polarization sum of Eq. (4.18) together with the coefficients of Eq. (4.21) and the normalization of Eq. (4.22) completely resembles the polarization sum of isotropic modified Maxwell theory based on the dimension-4 operator with the replacement . Therefore, I suspect that for the isotropic CPT-even Lorentz-violating photon sector including all higher-dimensional operators the polarization sum for each transverse mode has the same structure where is replaced by the general expansion of Eq. (IV.1) and by the corresponding modified dispersion relation. For vanishing Lorentz violation vanishes as well and . Then , which shows that in the standard case the normalization condition involving the 00-component of the energy-momentum tensor is automatically fulfilled.
Second, the terms with the coefficients and do not play a role when is contracted with a gauge-invariant quantity. This holds due to the Ward identity, which is still valid since the Lorentz-violating modification respects gauge invariance and no anomalies are expected to occur. Third, for vanishing Lorentz violation we have and, therefore, . The truncated polarization sum (meaning that all terms proportional to are dropped) then corresponds to the standard result PeskinSchroeder1995
Since the ansätze given by Eqs. (4.15), (4.18) are sufficient to describe the structure of the propagator and the polarization vectors, respectively, it is justified to take into account the timelike preferred spacetime direction only. There may be more directions, which are defined by the matrix . However they are not needed to understand the structure of the modification.
iv.2 Temporal part
Some of the general results presented above can be stated explicitly for the temporal part since they are not too lengthy. For this special case there are two isotropic dispersion laws (see Eq. (3.3)), i.e., they only depend on . Without loss of generality, the momentum three-vector can be chosen to point along the -axis. Then the transverse polarization vectors of Eq. (4.16) can be simplified to give (where one of the two possible signs is picked):
The perturbed and the spurious mode have different normalization factors that follow from Eq. (4.22) by inserting the corresponding dispersion law:
V Optical theorem at tree-level
The occurrence of spurious photon modes in the temporal and the mixed sector of the CPT-even modification based on the dimension-6 operator makes us curious about the validity of the optical theorem. The latter shall be studied in the current section where first of all the spatial case is considered. The optical theorem will be investigated based on a particular process: the scattering of a left-handed electron and a right-handed positron at tree-level (see Fig. 1). The calculation will be performed according to Schreck:2011ai ; Schreck:2013gma .
As long as no problems occur in the context of the optical theorem the imaginary part of the forward scattering amplitude must be related to the production cross-section of a modified photon from a left-handed electron and a right-handed positron. We will denote the matrix element of the latter process as . Note that it is not important, which process at tree-level is considered. In the proof no relationships will be employed that exclusively hold for this particular process. The only property, which is assumed, is the validity of the Ward identity. This is reasonable as the axial anomaly, which is linked to the chiral structure of quantum field theories, occurs at higher order of the electromagnetic coupling constant.
Now the forward scattering amplitude (left-hand side of Fig. 1) can be obtained with the standard Feynman rules for the fermion sector and the modified photon propagator. It reads as follows:
Here is the elementary charge, , , , and are standard Dirac spinors, with the standard Dirac matrices (for ), and is the unit matrix in spinor space. The kinematical variables used are shown in Fig. 1. The four-dimensional -function ensures total four-momentum conservation. The photon propagator with the propagator coefficients is taken from Eq. (4.19). The physical poles that appear in the scalar propagator part are treated with the ordinary -procedure. This means that the positive pole is shifted to the lower complex half-plane (meaning that an integration contour runs above the pole) and the negative one is shifted to the upper complex half-plane (where a contour runs below this pole).
v.1 Spatial part
For the spatial part of the modified theory considered the procedure used in Schreck:2011ai ; Schreck:2013gma does not fundamentally change. The denominator of the scalar propagator part is factorized with respect to the propagator poles. Terms of quadratic and higher order in the infinitesimal parameter are not taken into account. The photon propagator has two physical poles, where
is the positive one and corresponds to its negative counterpart. The scalar part of the propagator is then written in the following form:
Due to the -prescription the relation
with the principal value holds for the physical pole. The first part of Eq. (5.29) is purely real. The second part is imaginary and due to the -function it forces the zeroth four-momentum component to be equal to the respective physical photon frequency. The negative pole does not contribute to the imaginary part because of total four-momentum conservation. Furthermore
where follows from Eq. (4.22) by inserting the dispersion relation of Eq. (5.27). Using these results, the -integration in Eq. (V) can be done. Since the interest lies in the imaginary part, terms involving the principal value are not considered and is replaced by the photon frequency :
Terms that involve at least one four-momentum in the tensor structure of the propagator can be dropped, if the Ward identity is taken into account. Hence, the optical theorem at tree-level is valid for the spatial sector as expected.
v.2 Temporal part
The temporal part is characterized by two distinct dispersion relations according to Eq. (3.3). To make the following calculations more transparent they will be written as follows:
The first of these is a perturbation of the standard dispersion law whereas this is not the case for the second. Therefore the second can be considered as spurious for momenta that are much smaller than the Planck scale. Nevertheless there is no reason why it formally should not be taken into account in the optical theorem. Although it is considered as spurious it is, indeed, a transverse dispersion law (see the discussion in Sec. III.1). When is not discarded, the structure of the temporal sector is reminiscent of the structure of a birefringent theory. In Schreck:2013gma a birefringent sector of modified Maxwell theory was considered that is based on the dimension-4 operator. If both and are assumed to contribute to the imaginary part of the forward scattering amplitude the calculation can be performed analogously to how this was done in the latter reference. The only difference is that each transverse dispersion relation is linked to a separate polarization sum according to Eq. (4.17) and not only to one of the two contributions of the sum. This has to do with the twofold degeneracy of each dispersion law (cf. the last paragraph of Sec. III).
The scalar part of the modified photon propagator then has four different poles. Two of them are given by Eq. (5.32) and the other two by their respective negative counterparts. The denominator of the scalar part is factorized with respect to these poles and the -prescription is applied again:
Then Eq. (5.29) can be used for each of the positive poles. The negative ones do not play a role due to four-momentum conservation. This results in the following contributions to the imaginary part, where a factor of is omitted:
According to the previous section and the discussion in Schreck:2013gma , for the optical theorem to be valid these expressions have to correspond to the following results where and