BUCMP/02-03

UMSNH-Phys/02-6

Two-loop self-dual Euler-Heisenberg Lagrangians (I):

Real part and helicity amplitudes

Gerald V. Dunne

[1.5ex] Department of Physics

University of Connecticut

Storrs CT 06269, USA

Christian Schubert

[1.5ex] Instituto de Física y Matemáticas

Universidad Michoacana de San Nicolás de Hidalgo

Apdo. Postal 2-82

C.P. 58040, Morelia, Michoacán, México

Center for Mathematical Physics, Mathematics Department

Boston University, Boston, MA 02215, USA

California Institute for Physics and Astrophysics

366 Cambridge Ave., Palo Alto, CA 94306, USA

Abstract

We show that, for both scalar and spinor QED, the two-loop Euler-Heisenberg effective Lagrangian for a constant Euclidean self-dual background has an extremely simple closed-form expression in terms of the digamma function. Moreover, the scalar and spinor QED effective Lagrangians are very similar to one another. These results are dramatic simplifications compared to the results for other backgrounds. We apply them to a calculation of the low energy limits of the two-loop massive N-photon ‘all +’ helicity amplitudes. The simplicity of our results can be related to the connection between self-duality, helicity and supersymmetry.

## 1 Introduction: QED and QCD in constant fields

The Euler-Heisenberg Lagrangian, one of the earliest results in quantum electrodynamics [1, 2], describes the complete one-loop amplitude involving a spinor loop interacting non-perturbatively with a constant background electromagnetic field. Euler and Heisenberg found the following well-known integral representation for this effective Lagrangian:

Here denotes the (Euclidean) propertime of the loop fermion, and are related to the two invariants of the Maxwell field by . The subtractions of the terms of zeroeth and second order in corresponds to on-shell renormalization.

The Euler-Heisenberg Lagrangian contains the information on the low – enery limit of the one–loop –photon amplitudes for any . Moreover, it does so in a form which is extremely convenient for the study at low energies of nonlinear QED effects [3] such as photon–photon scattering [1], photon dispersion [4, 5], and photon splitting [6, 5, 7, 8]. For scalar QED, an analogous result was obtained by Schwinger [9]:

The Lagrangians (LABEL:L1spinren) and (LABEL:L1scalren) are real for a purely magnetic field, while in the presence of an electric field there is an absorptive part, indicating the process of electron–positron (resp. scalar–antiscalar) pair creation by the field [9].

The first radiative corrections to these Lagrangians, describing the effect of an additional photon exchange in the loop, were obtained in the seventies by Ritus [10, 11, 12]. Using the exact propagators in a constant field found by Fock [13] and Schwinger [9], and a proper-time cutoff as the UV regulator, Ritus obtained the corresponding two-loop effective Lagrangians in terms of certain two-parameter integrals. Similar two-parameter integral representations for where obtained later by other authors, using either proper-time [14, 15] or dimensional regularisation [16, 17]. Unfortunately, all of these double parameter integral representations are quite complicated, so that it is much more difficult to study the weak- and strong-field expansions at two-loops than at one-loop. Even more difficult becomes the analysis of the imaginary part of the effective action [18]. This is true even for the special cases where the background is purely magnetic or purely electric [11, 12].

However, the magnetic/electric cases are not the simplest ones
which one can study in this context. As we will see in the following,
the constant field background which leads to maximal simplification
is the (Euclidean) self-dual one, given by
^{1}^{1}1This Euclidean self-duality condition should not be
confused with the similar condition which can
be realized in Minkowski space and could also be called
‘self-duality’. See [19] for a discussion of these
various dualities.

(1.3) |

This is because for such a field the square of the field strength tensor is proportional to the Lorentz identity (see eq.(2.2) below) so that the inevitable breaking of the Lorentz invariance by the background field is in some sense minimized. In Minkowski space the self-duality condition requires either E or B to be complex. However, this does not imply that such backgrounds are devoid of physical meaning. Rather, they should be interpreted in terms of helicity projections [20]. And indeed, it will be seen below that it is quite straightforward to extract, from the effective action for such a self-dual field, the low energy limit of the two-loop amplitude for the scattering of N photons with all equal helicities. In addition, we see at least four more good reasons for studying this particular self-dual case in detail:

First, a detailed comparison at the one-loop level shows that the self-dual case with real and complex (corresponding to a real self-dual field strength , where , and called the ‘magnetic’ case in the following) leads to an effective Lagrangian whose properties are only marginally different from the ones of the magnetic Euler-Heisenberg Lagrangian, while the case with a real electric and complex magnetic field (corresponding to imaginary, and called the ‘electric’ case in the following) is a very good analogue of the electric case. This leads us to expect that the study of the self-dual case at higher loops may yield useful information on the generic properties of Euler-Heisenberg Lagrangians.

Second, many quantities of physical interest in quantum electrodynamics are computable in Euclidean space. This includes the renormalization constants, and in particular the QED –function.

Third, the constant self-dual Euclidean backgrounds that we consider here for QED generalize in a very simple way to the case of quasi-abelian self-dual constant backgrounds in QCD. Such backgrounds have been studied extensively in QCD as they have the special property [21] that among the covariantly constant gluon backgrounds, only the self-dual quasi-abelian background is stable (at one-loop) under fluctuations. This has led to extensive studies of quark and gluon loops in such a background [21, 22, 23, 24]. Also at one-loop, much is known about QCD in the presence of arbitrary covariantly constant background fields. One-loop results along the lines of the Euler-Heisenberg formulas (LABEL:L1spinren) and (LABEL:L1scalren) can be found in [25, 26]. The purely chromomagnetic covariantly constant background is unstable [27, 28, 21]. The renormalization of these one-loop results in pure QCD is complicated by infrared problems, problems which become even worse at the two-loop level [29].

A fourth motivation for studying constant quasi-abelian self-dual backgrounds is that they could provide useful information about effective Lagrangians in other self-dual backgrounds, in particular instantons [30, 31, 32].

As our main result in this paper, we will show that in the self-dual case, at two-loop, all parameter integrals can be done in closed form, yielding the following simple expressions for the two-loop scalar and spinor QED effective Lagrangians:

(1.4) | |||||

(1.5) |

Here we have defined the convenient dimensionless parameter

(1.6) |

where , as well as the important function

(1.7) |

with being the digamma function . The scalar QED formula (1.4) was already presented in [33].

The extreme simplicity of these results, and the obvious similarity between the two-loop effective Lagrangians in the scalar and spinor cases, provides a new motivation for studying self-dual backgrounds. Our results provide a new example of the well-known connections among self-duality, helicity and supersymmetry. Self-dual fields have definite helicity [20], and are closely related to supersymmetry [30, 34, 35]. One consequence of this connection is that there exist remarkably simple formulas for loop amplitudes when the external fields have all (or almost all) helicities being equal. Such simplifications have been known for a long time in massless QCD at the tree level [36]. More recently they have been studied extensively at the one- and two loop level, in abelian [37, 38, 39] as well as in nonabelian [40, 41, 42] gauge theory. The close interplay been self-duality, supersymmetry and integrability has also been explored in this context. The simplicity of the structure of the QED/QCD amplitudes with all (almost all) helicities alike is thought to be deeply related to the integrability properties of self-dual Yang-Mills fields [43, 44]. In the present paper the connection between self-duality and helicity will be used to obtain, from the above results for the self-dual two-loop effective Lagrangians, the low energy limits of the QED - photon amplitudes with all helicities alike.

The outline of the paper is as follows. In section 2 we write down the one-loop self-dual Euler-Heisenberg Lagrangians for scalar and spinor QED, as well as for the ‘quasi-abelian’ special case in QCD. In section 3 we compute the two-loop corrections to these Lagrangians for scalar and spinor QED, using the ‘string-inspired’ worldline formalism [45, 46] along the lines of [47, 48, 15, 16]. This formalism is based on the representation of effective actions in terms of first-quantized particle path integrals, and has turned out to be highly convenient for computations involving constant external fields [47, 49, 50, 51, 52, 15, 8, 16] (see [53] for a review). In section 4 we comment on the special properties of self-dual fields with respect to helicity and supersymmetry, and we use some of these properties to explain the similarity of our two-loop scalar and spinor QED results. We then proceed in section 5 to an explicit calculation of the low energy limits of the massive all ‘+’ helicity N photon amplitudes, at one and two loops, in scalar and spinor QED. Section 6 contains a summary, and some possible future directions of work. In an accompanying paper [54] we study the weak- and strong-field expansions of these two-loop effective Lagrangians, and use them to test the techniques of Borel summation, and the associated analytic continuation properties and structure of the imaginary part of the two-loop self-dual Euler-Heisenberg Lagrangians for the ‘electric’ case.

## 2 One loop self–dual Euler-Heisenberg Lagrangians

### 2.1 Scalar QED

For scalar QED, the (unrenormalized) one-loop effective Lagrangian in a constant background is [53]

(2.1) |

Here is the proper-time variable for the loop scalar, and is the spacetime dimension. We would like to evaluate this integral for the case of a self-dual field, . The self-duality condition implies that

(2.2) |

where , and denotes the identity matrix in Lorentz space. Therefore

(2.3) |

Renormalization involves subtracting the free field contribution, and also a charge renormalization, corresponding to subtracting the logarithmically divergent term. This leads to the following renormalized one-loop effective Lagrangian:

(2.4) |

We remark that this integral can be expressed in terms of the function and its integral function:

(2.5) |

where the function is defined as follows:

(2.6) |

This one-loop result (2.5) may equivalently be expressed in terms of the Hurwitz zeta function (see, e.g., [14, 55]), but we prefer to use this representation here, since the two-loop result is a simple function of .

### 2.2 Spinor QED

For spinor QED, the unrenormalized one-loop effective Lagrangian in a constant background is

(2.7) |

For a self-dual field

(2.8) |

After renormalization, this leads to the following effective Lagrangian:

(2.9) |

### 2.3 Qcd

In the nonabelian case, the natural notion of a constant field strength is covariant constancy, i.e. . This does not imply that is a constant matrix, so that further assumptions need to be made to arrive at Euler-Heisenberg type formulas. The simplest such case is the ‘quasi-abelian’ one, where , and therefore also , are assumed to point in a fixed direction in colour space:

(2.10) |

The computation of the one-loop effective action with such a field is identical to the abelian one for the scalar and spinor loop cases, since the colour degree of freedom manifests itself only in a global colour trace. Thus for the self-dual case one obtains (compare (2.1),(2.7))

(2.11) | |||||

(2.12) |

For the gluon loop one finds, in this quasi-abelian constant field case, the result [26, 55, 15]

In these formulas, denotes the Lorentz trace, the colour trace, and we defined LABEL:1lqcd) can be written as . For the self-dual case, (

Let us specialize further to the case where all particles are in the adjoint representation and massless. We can then combine the contribution of (real) scalars, (Weyl) spinors, the gauge boson and its ghost (which gives minus the contribution of a complex scalar) into (putting )

For and we recover the well-known fact that the one-loop self-dual effective action vanishes for SYM theory [34, 20, 56].

## 3 Two loop self–dual Euler-Heisenberg Lagrangians

### 3.1 Scalar QED

In [15, 16] the two-loop Euler-Heisenberg Lagrangian in (Euclidean) Scalar QED was obtained in terms of the following fourfold parameter integral,

(3.1) | |||||

Here and represent the scalar and photon proper-times, and the endpoints of the photon insertion moving around the scalar loop. and are given by

(3.2) |

They are expressed in terms of the worldline Green’s function and its first and second derivatives [15]:

(3.3) |

These are Lorentz matrices, and the above formulas should be understood as power series in the field strength tensor . The scalar function is given by . Due to the translation invariance of those Green’s functions one of the integrations is redundant, so that we will set in the following.

Removing the second derivative by a partial integration with respect to or one can obtain the equivalent integral

(3.4) | |||||

We would again like to evaluate this integral for the case of a self-dual field. The worldline correlators (3.3) for such a field simplify to the following [57]:

(3.5) |

where . The first determinant factor is already known from the one-loop calculation, eq. (2.3). The Lorentz matrix , being an even function of , for this background becomes scalar:

(3.6) |

Therefore in the self-dual case the integrals in eqs.(3.1),(3.4) can be done trivially. In the same way as in the magnetic field case [15, 16], further simplification is achieved by taking the following linear combination of the representations (3.1) and (3.4),

(3.7) |

After the usual rescaling to the unit circle,
,
and setting , we end up with the
following twofold integral,
^{2}^{2}2Note that the - term contained in
has been deleted, since here the
photon proper-time integral is a
tadpole type integral
which vanishes in dimensional regularization.

(3.8) |

where

(3.9) |

(). We are only interested in terms of order and higher in this Lagrangian, since the term contributes only to photon wave function renormalization. For those terms the final - integration is finite, so that in we can set immediately. The same is not possible for , since this integral for has a divergence at the points where the two photon end points become coincident, . For the calculation of this integral we split it in the following way,

Of these four integrals the first one is of order O and can be omitted. The second and third ones are equal and elementary,

The fourth integral can be calculated by transforming to the variable , and using formula 22.10.11 of [58] (see [59] for a general technique for computing this type of integral). The result is

(3.12) |

with the Euler Beta function. Expanding in we find

(3.13) | |||||

We have written this result for the Lagrangian in terms of bare quantities, since it still requires renormalization. Mass renormalization requires us to subtract a term , where is the one-loop Euler-Heisenberg Lagrangian (2.4), and the one-loop mass shift, both calculated in dimensional regularization. The mass shift is

(3.14) |

Noting that

(3.15) |

we perform a partial integration on the first term on the right hand side of (3.13). After this the pole term in it takes the same form as the pole of the mass shift term above. Subtracting the mass shift term we end up with the following expression for the renormalized Lagrangian,

This can also be written as

Note that, since we have not yet performed the photon wave function renormalization, this formula is correct only beginning from .

For the calculation of this integral, we go back to the regularized version, eqs. (3.8) – (3.12), and note that it involves essentially only the integral

(3.18) |

However, to be able to take the limit we instead introduce

(3.19) |

Including the mass renormalization term, and again disregarding terms, we can then write

where we have finally subtracted out the terms of order , .

### 3.2 Spinor QED

The calculations for the spinor QED case are very similar, so we content ourselves primarily with a presentation of the results.

For the two-loop Euler-Heisenberg Lagrangian in Euclidean spinor QED, the worldline formalism leads to the following parameter integral, which is completely analogous to (3.4) [15, 16]:

(3.21) | |||||

The bosonic worldline Green’s function is given in (3.3), while the fermionic one is:

(3.22) |

where . For a self-dual background simplifies to

(3.23) |

The calculation, together with the charge and mass renormalization, proceed in a manner analogous to the two-loop scalar QED calculation in the previous section (for details of the mass renormalization, see [15]). After similar manipulations we find that the on-shell renormalized effective Lagrangian can be written as the following proper-time integral:

Here we have also subtracted the zero-field term and done the charge renormalization.

As in the scalar case, this proper-time integral can be done in closed-form. After manipulations similar to those in the scalar case, we find the final result (1.5),

where is the same function that was defined before in (1.7). This result (1.5) is remarkably similar to the two-loop scalar QED result (1.4). We find it interesting that the spinor and scalar QED results for a constant selfdual background can each be written in such a simple form in terms of the same function . This fact is discussed in the next section.

## 4 Self-duality, helicity and supersymmetry at one- and two-loops

In this section we explain why the renormalized scalar and spinor effective Lagrangians in a self-dual background are so similar, at both one- and two-loop. The self-duality of the background has the effect that both the scalar and spinor cases are dramatically simplified from the results for a general constant background. But, on top of these simplifications, the scalar and spinor results (1.4) and (1.5) are remarkably similar to one another, each being expressed as a simple function of . The key to understanding this correspondence is the connection between self-duality, helicity and supersymmetry. It is well known that self-dual gauge fields are closely related to supersymmetry, and that they correspond to helicity eigenstates [30, 34, 20, 35]. In this section we explain how these connections are manifest in our exact effective action results at both one- and two-loops.

Consider the one-loop QED case. Since , we compare the one-loop renormalized effective Lagrangians (2.4) and (2.9) for the scalar and spinor cases, respectively, and find that

(4.1) |

Note that this proportionality between the scalar and spinor effective Lagrangians is only true for a self-dual background. Furthermore, the relation (4.1) holds for the renormalized effective Lagrangians, not for the unrenormalized ones.

This proportionality between the one-loop scalar and spinor effective Lagrangians reflects a well-known supersymmetry of the self-dual background [30, 34, 20, 35]. The basic relation between self-dual fields and helicity can be traced to the following identity (we work in Euclidean space):

(4.2) |

which holds provided the field strength is self-dual. This fact has the consequence [30, 20] that the bosonic and fermionic operators

(4.3) | |||||

(4.4) |

have identical spectra, but the fermionic case has a four-fold multiplicity (from the spinor degrees of freedom). This is due to a quantum mechanical supersymmetry [34] relating the bosonic and fermionic operators in (4.3) and (4.4). Therefore,

(4.5) |

which is precisely the one-loop relation (4.1).

Another equivalent way to see this is to note that because of the helicity/self-duality relation (4.2), the spinor propagator in a self-dual background can be expressed in a simple manner in terms of the scalar propagator in the same background [35]:

(4.6) |

where the scalar and spinor propagators are

(4.7) |

This relation between the spinor and scalar propagators in a self-dual background provides another perspective on the relation (4.1). The spinor induced current can be related to the scalar induced current since,

(4.8) | |||||

where in the last step we have used (4.6). This current relation agrees with the one-loop effective Lagrangian relation (4.1).

At the two-loop level, the supersymmetry relation (4.1) no longer holds, as can be seen by comparing the two-loop scalar and spinor results (1.4) and (1.5), respectively. Namely,

(4.9) |

This non-vanishing is because the two-loop effective Lagrangians are not simply logarithms of determinants. Nevertheless, it is interesting to see that each of the scalar and spinor two-loop effective Lagrangians can be written in such a simple form in terms of the same function . To understand why this is the case, let us reconsider this computation in terms of more standard field theory techniques. Using Ritus’s approach [12] the two-loop effective Lagrangians can be written as [12]:

(4.10) | |||||

(4.11) |

where and are the position space spinor and scalar propagators in the presence of the background field, and denotes the internal photon propagator. We have not written tadpole terms, as these do not contribute after renormalization. These expressions (4.10) and (4.11) are valid for a general background. Now, if we specialize to a self-dual background, then the spinor and scalar propagators become related as in (4.6), which means that the Dirac traces can be done in the spinor case, leading to an expression in terms of the scalar propagator:

(4.12) |

Thus, comparing with the scalar two-loop effective Lagrangian (4.11), we see that the spinor two-loop effective Lagrangian for a self-dual background involves the same structures but with different coefficients. This is reflected directly in our explicit closed-form two-loop expressions (1.4) and (1.5).

To make this connection more precise, we consider a general linear combination of these two structures: