# Non-Perturbative One-Loop Effective Action

for QED with Yukawa Couplings

###### Abstract

We derive the one-loop effective action for scalar, pseudoscalar, and electromagnetic fields coupled to a Dirac fermion in an extension of QED with Yukawa couplings. Using the Schwinger proper-time formalism and zeta-function regularization, we calculate the full non-perturbative effective action to one loop in the constant background field approximation. Our result is non-perturbative in the external fields, and goes beyond existing results in the literature which treat only the first non-trivial order involving the pseudoscalar. The result has an even and odd part, which are related to the modulus and phase of the fermion functional determinant. The even contribution to the effective action involves the modulus of the effective Yukawa couplings and is invariant under global chiral transformations while the odd contribution is proportional to the angle between the scalar and pseudoscalar couplings. In different limits the effective action reduces either to the Euler-Heisenberg effective action or the Coleman-Weinberg potential. We also comment on the relationship between the odd part of the effective action and the chiral anomaly in QED.

###### pacs:

12.20.-m, 11.30.Rd, 12.20.Ds, 11.10.Gh## I Introduction

The effective action provides rich insight into the low energy regime of an underlying quantum field theory. The earliest example of a full non-perturbative effective action is the Euler-Heisenberg effective action, which describes the non-linear quantum corrections to the classical Maxwell theory, in the one-loop approximation Heisenberg:1935qt ; Weisskopf:1996bu ; doi:10.1142/S0217751X1430052X . Schwinger’s calculation of the same effective Lagrangian makes use of Fock’s proper-time formalism and re-frames the pair production rate as a signal of the instability of the QED vacuum Fock ; Schwinger . The pair production rate can be read off from the imaginary part of the Euler-Heisenberg lagrangian, which vanishes to all orders in perturbation theory. Hence, the non-perturbative nature of the calculation captures essential physical phenomena, supplementing perturbative studies Dunne1 ; Dittrich:2000zu .

The Euler-Heisenberg result, which is obtained assuming the background electromagnetic field is constant (i.e. slowly varying), has been extended to other solvable backgrounds PhysRevD.52.2422 , to supersymmetric QED, to second order in the loop expansion Clark:1987et ; Kuzenko:2003qg ; Kuzenko:2007cg ; Dunne1 ; Dunne:2001pp , and to QED in gravitational backgrounds Fucci:2009je ; PhysRevD.22.343 ; Bastianelli:2008cu ; PhysRevD.50.909 . For a review of the original Euler-Heisenberg effective action and its extensions, as well as its historical development, see Dunne1 ; Dunne:2012vv .

The Fock-Schwinger proper-time approach and its generalizations, which appear in both perturbative and non-perturbative calculations of effective Lagrangians, are especially useful because they are symmetry-preserving and can be applied to theories involving the totally antisymmetric tensor McKeonSherry . The proper-time formalism has been generalized to heat-kernel, zeta-function, and operator regularization, which provide powerful techniques for computing and regularizing the formal determinants and inverses of functional operators dewitt1965dynamical ; Dunne2 ; Reuter:1984kw ; Mckeon:1987ea ; MCARTHUR1997525 .

The standard procedure in operator regularization begins with the heat-kernel representation for a positive operator ,

(1) |

At one-loop order, deriving the effective action involves calculating , where is an operator which appears in the underlying theory and involves the background fields. For cases when is not Hermitian, let alone positive, the fermion determinant must be split into its modulus and phase, which generate “even” and “odd” contributions to the effective action, respectively McKeonSherry . In the present work, we find a much simpler way of calculating the even and odd contributions to the effective action which circumvents the need for computing the phase of the functional determinant directly.

In the perturbative regime, proper-time techniques have been used to compute the radiatively induced effect of adding Lorentz- and CPT-violation to QED Sitenko:2001iw ; Sitenko:2002fy ; PhysRevD.89.045005 ; Borges:2016uwl , as well as the effective action for the Yukawa model in curved spacetime Toms:2018wpy . In the non-perturbative regime, heat-kernel methods were used to obtain the world-line path integral for fermions with general scalar, pseudoscalar, and vector couplings DHoker:1995aat ; DHoker:1995uyv . While world-line path integrals are in the same spirit, the full closed-form effective action for QED with Yukawa couplings has not yet been derived. In the present paper, we provide a simple derivation of the effective action for fermions in the one-loop and constant background field approximations. Our result is non-perturbative in the background scalar, pseudoscalar, and electromagnetic fields. The even portion of the effective action is similar to the Euler-Heisenberg effective action, except the fermion mass is modified by the Yukawa couplings. The odd portion of the effective action is proportional to the CP-odd Lorentz scalar and the angle between the effective scalar and pseudoscalar Yukawa terms.

## Ii The model

We consider a simple model consisting of a Dirac fermion coupled to a background scalar , pseudoscalar , and gauge field , with Lagrangian

(2) |

In the signature, the QED and Yukawa Lagrangians are

(3a) | |||||

(3b) |

where is the field strength, the covariant derivative is , and and are real coupling constants. The scalar potential also contains possible mass terms for the scalar fields, and is the gauge-fixing parameter. Throughout the following we will use the Dirac slash notation . With our metric signature, the gamma matrices satisfy

(4) |

and we adopt the following definition for ,

(5) |

where is totally antisymmetric with . The background fields are assumed to be Hermitian, , , , and we assume a representation for the gamma matrices such that

(6) |

With these Hermiticity conditions, one can easily check that the Lagrangian density (2) is real.

In the constant field approximation, is essentially treated as a VEV which simply modifies the fermion mass . Furthermore, at the classical level the pseudoscalar Yukawa coupling can be removed by transforming and , for a suitably chosen constant . At the quantum level this is no longer the case, and as we will see, the pseudoscalar coupling gives rise to the odd part of the effective action. We will revisit the role of the parameter in more detail in Subsection III.2.

## Iii One-Loop Effective Potential

The effective action is formally defined to be

(7) |

which, for the Lagrangian (2) and in the one-loop approximation, reads

(8) |

where is a trace over Dirac indices as well as spacetime.

Due to the presence of the pseudoscalar coupling, the operator appearing in the theory is not Hermitian, and the standard procedure for computing functional determinants does not apply. Rather than computing the fermion determinant directly, we instead compute its contribution to the pseudoscalar current,

(9) |

This approach circumvents the more delicate splitting of the fermion determinant into its modulus and phase. Now we multiply the numerator and denominator by ,

(10) |

assuming and are constant background fields. The denominator in (10) is positive, and hence can be represented by a proper-time integral. Taking the limit in (1),

(11) |

where we have used the fact that times an odd number of is traceless. The current splits into two parts, which we call even and odd,

(12a) | ||||

(12b) |

The remaining step in the calculation involves evaluating the traces

(13) |

### iii.1 Coincidence limit of the fermion propagator

There are various methods for calculating (13) available in the literature, for instance Schwinger ; Schwartz:2013pla ; ItzyksonZuber ; Bagrov:1529647 . For completeness, we review the calculation, loosely following the procedure found in chapter 4 of ItzyksonZuber , evaluating (13) directly rather than taking the coincident limit of the full propagator. Those familiar with the calculation can proceed to Subsection III.2.

We begin by splitting into symmetric and antisymmetric parts,

(14) |

Together with , this yields the identity

(15) |

Since is constant and hence commutes with , the exponential factorizes,

(16) |

Using the identity

one can easily show that

(18) |

where . Defining , we have

(19) | |||||

We are not interested in the traceless terms in

(20) |

and because , we can disregard the odd terms in the expansion (19). For the even terms, it is useful to expand the powers of into parts proportional to the identity and parts proportional to . To demonstrate, the first four powers of are

(21) | |||||

In order to re-sum this series we make an explicit choice for the form of . So long as the electric and magnetic fields are not perpendicular, we can always Lorentz-transform to a frame where they are parallel. Hence, without loss of generality, we let the electric and magnetic fields point in the -direction. For and , where and are space-time constants, the Lorentz-invariant combinations become

(22) |

Inverting these equations gives and in terms of the general Lorentz invariant quantities and ,

(23a) | |||||

(23b) |

Once expressed in terms of and , the expansion (19) can be easily re-summed,

(24) |

The odd terms have a simple relation to the even terms but for our present purposes they are not necessary and we shall omit them.

Continuing with our calculation of (16), we need to evaluate

(25) |

Analytically continuing , and introducing the operator , we can identify the proper-time Hamiltonian

(26) |

The classical gauge field configuration is a function of the space-time coordinates , which satisfy canonical commutation relations,

(27) |

Up to a gauge transformation, the nonzero components of the electromagnetic potential are . The Hamiltonian is thus

(28) | |||||

We now use the Baker-Campbell-Hausdorff formula,

(29) |

where and . Using , we have the following,

(30) | |||||

where all successive terms in the Hausdorff expansion are zero. Thus,

(31) |

Similarly, we find

(32) |

The original matrix element then factorizes as follows,

(33) |

where we have defined . We begin with , inserting a complete set of momentum states,

(34) | |||||

When we exponentiate a Hamiltonian of this form, we get

(35) |

Making use of this identity and inserting another complete set of states, we have

(36) | |||||

Integrating over and , we have

(37) | |||||

Integrating over , we get

(38) | |||||

We recognize this as a harmonic oscillator with Hamiltonian

(39) |

with energy spectrum

(40) |

The matrix element we want to calculate is thus

(41) | |||||

The sum is clearly the Taylor expansion of

(42) |

so in the end we have

(43) |

The case for is similar, and we will not repeat the derivation here. We note, however, that we are free to choose the positive or negative square root when defining the “Landau” frequencies and . By setting , the result matches the free-field propagator in the limits . With this choice, the matrix element is

(44) |

Finally, we return to obtain our result,

(45) |

### iii.2 Effective lagrangian

Putting the pieces together, the pseudoscalar current becomes

(47a) | |||||

(47b) |

Integrating with respect to and multiplying by , we find the even and odd parts of the effective Lagrangian: , where

(48a) | ||||

(48b) |

This is our main result.

It is important to note that we have captured the full non-perturbative effects of the pseudoscalar coupling. Though the first-order term from can be found in Schwinger , our result is valid *to all orders* in . The closed-form Lagrangian (48b) also exhibits some interesting features. Most strikingly, only appears, with no higher order corrections past . Hence, any couplings involving higher odd powers of (if they appear in the effective theory at all) must necessarily involve derivatives of or . In addition, it is interesting to note that for very large arguments the arctangent is approximately constant and tends to . Hence, when (for vanishing ) the ratio , the term is nearly a total derivative.

We now return to the question of chiral invariance. While the even portion of the effective action clearly retains a global chiral symmetry, the odd portion is in fact related to the parameter introduced in Section II. To be more explicit, the pseudoscalar term in the QED-Yukawa Lagrangian can be removed with a chiral rotation where the transformation parameter satisfies

(49) |

which allows us to rewrite the odd portion of the effective action as

(50) |

However, the parameter is spacetime-independent only if and are also spacetime-independent. As the effective action (48) represents the leading contribution in a derivative (momentum) expansion, we are treating and not as constants but as slowly-varying fields. From this point of view, we should treat as a local parameter which fails to leave the kinetic term of (3a) invariant. Nonetheless, the identification of in the odd portion of the effective action sheds light on chiral invariance in the effective theory. The even part of the effective action depends on the “modulus” , whereas the odd part depends on the angle itself between and .

A note about passing from the pseudoscalar current (47) to the effective Lagrangian (48). In the process of integration we have the freedom to add to the Lagrangian an arbitrary functional that does not depend on . However, we can just as easily vary (8) with respect to or and obtain (48a) and (48b) up to a total derivative. Hence, we are free to add some function of only, which by dimensional grounds must either be a correction to the cosmological constant, the free photon term , or . The first two can be subtracted off with the appropriate counter-terms, and the third is a total derivative. Hence, the expressions (48a) and (48b) hold without loss of generality.

To conclude, we have derived the even and odd parts of the full non-perturbative effective potential for QED with general Yukawa couplings, valid to all orders in the background scalar, pseudoscalar, and electromagnetic fields. More precisely, it is the zeroth-order result in the derivative expansion of the full effective Lagrangian. The even portion is simply the Euler-Heisenberg effective action but with the formal replacement which is expected from the Dirac structure of the Yukawa couplings. The even part only depends on , and hence generates graphs with even numbers of pseudoscalar vertices. The odd portion is (somewhat surprisingly) proportional to with no higher-order corrections appearing.

In the limit , the odd portion vanishes and we exactly recover the Euler-Heisenberg Lagrangian. Similarly, if we let , the odd portion vanishes, but the even portion becomes

(51) |

The bad behavior of the integral as reflects an ultraviolet divergence. As we will see, the divergence can be handled with appropriate counter-terms.

## Iv Weak Field Limit and Renormalization

In this section we consider the perturbative expansion of (48) in powers of . We begin with the even part, given by (48a). Grouping by powers of , and using (23), the weak-field expansion becomes

(52b) | |||||

which is manifestly gauge-invariant. For ease of notation we have introduced

(53) |

The first two integrals (52b) are badly behaved as , and require regularization. This can be achieved by cutting off the lower bound of the integral at a small positive number (making use of the incomplete Gamma function), or by analytic continuation of the complete Gamma function. We take the latter approach. Despite its resemblance to dimensional regularization, this regularization procedure does not analytically continue the number of spacetime dimensions, but rather the power of in the denominator of the integrand, which is not physical. Making use of the integral representation of the Gamma function,

(54) |

we can compute the integrals above. The result is

(55) | |||||

where is the Euler-Mascheroni constant and the limit is assumed. Now we proceed in the scheme, adding two counter terms

(56a) | |||||

(56b) |

where we have introduced the renormalization scale to ensure the counter terms have the correct dimension. The first counter term corresponds to the renormalization of the cosmological constant and the scalar potential ; specifically, the and couplings. The second counter term corresponds to the renormalization of the free photon term and is related to the vacuum polarization. The limit of the sum is finite,

(57) | |||||