Group invariants for Feynman diagrams

It is well-known that the symmetry group of a Feynman diagram can give important information on possible strategies for its evaluation, and the mathematical objects that will be involved. Motivated by ongoing work on multi-loop multi-photon amplitudes in quantum electrodynamics, here I will discuss the usefulness of introducing a polynomial basis of invariants of the symmetry group of a diagram in Feynman-Schwinger parameter space.


Introduction: Schwinger parameter representation of Feynman diagrams
The most universal approach to the calculation of Feynman diagrams uses Feynman-Schwinger parameters x i , introduced through the exponentiation of the (Euclidean) scalar propagator, For scalar diagrams, one finds the following universal structure for an arbitrary graph G with n internal lines and l loops in D dimensions: U and F are polynomials in the x i called the first and second Symanzyk (graph) polynomials.There exist graphical methods for their construction.

031.1
For more general theories (involving not only scalar particles) the same graph will involve, in addition to these two polynomials, also a numerator polynomial N (x 1 , . . ., x n ).Such numerator polynomials can become extremely large, and in the present talk I would like to point out the universal option of rewriting them as polynomials in a basis of invariants of the symmetry group of the graph.After introducing the symmetries of Feynman diagrams and some related facts of invariant theory in section 2, sections 3 to 8 are devoted to the discussion and motivation of our main example, the three-loop effective Lagrangian in two-dimensional QED, where we have found this procedure to lead to significantly more manageable expressions.We come back to the group-theoretical aspects of this calculation in section 9, before summarizing our findings in sections 10 and 11.

Symmetries of Feynman diagrams
Many Feynman diagrams possess a non-trivial symmetry group, generated by interchanges of the internal lines that leave the topology of the graph unchanged. 1Then all its graph polynomials must be invariant under the natural action of the group on the set of polynomials IR[x 1 , . . ., x n ], g.P(X ) ≡ P(g.X ).
A nice example is the l − l oop banana graph shown in Fig. 1.The zero locus of the denominator of the integral defines a singular family of (l 1)-fold Calabi-Yau hypersurfaces M s as Due to standard arguments, see e.g.[31], M s is a complex Kähler manifold with trivial canonical class K = 0, hence a Calabi-Yau space.The first fact follows by the definition of M s as hypersurface in projective space P l and the second as for a homogeneous polynomial P l of degree deg(P ) in P l the canonical class is given in terms of the hyperplane class H and deg(P ) = (l+1).Note that given the scaling of (2.3) this degree makes the integrand of (2.1) well defined under the C ⇤ scaling of the homogenous coordinates defining P l .Embedded in P l the hypersurface is a singular Calabi-Yau space.Due to the Batyrev construction there is a canonical resolution of these singularities to define a smooth Calabi-Yau family, which we discuss next following [19,7,21,28].A Calabi-Yau manifold M of complex dimension n = l 1 has two characteristic global di↵erential forms.Since it is Kähler it has a Kähler (1, 1)-form !defining its Kähler-or symplectic structure deformations space.The triviality of the canonical class implies the existence of an unique holomorphic (n, 0)-form that plays a crucial role in the description of the complex structure deformations space of M .

Calabi-Yau hypersurfaces in toric ambient spaces
First we define a Newton polynomial P l as (2.5) The exponents of each monomial of P l , w.r.t. to the coordinates x i , i = 1, . . ., l + 1, define a point in a lattice Z l+1 .The convex hull of all these points in the natural embedding of Z l+1 ⇢ R l+1 defines an l-dimensional lattice polyhedron.The dimension is reduced due to the homogeneity of P l and we denote the polyhedron10 that lies in the induced lattice Z l ⇢ R l by l .When all the masses M i are equal, this graph has full permutation symmetry in all the internal lines, so the symmetry group is S l+1 , and its graph polynomials must be symmetric functions of x 1 , . . ., x l+1 .As is well-known, this implies that they can be rewritten as polynomials in the elementary symmetric polynomials S 1 , . . ., S n .
Perhaps less known is that this generalizes to the case of a general symmetry group as follows [1]: Theorem 1: Let G be a finite group and let Γ be an n−dimensional (real) representation.
1.There exist n = dim(Γ ) algebraically invariant polynomials P 1 , • • • , P n , called the primitive invariants, such that the Jacobian the subalgebra of polynomial invariants generated by the primitive invariants.

There exist
In particular this means that any invariant There exist computer algebra systems for the computation of P 1 , . . ., P n such as SINGULAR [2].

The Euler-Heisenberg Lagrangian at one loop
In 1936 Heisenberg and Euler obtained their famous representation of the one-loop QED effective Lagrangian in a constant field ("Euler-Heisenberg Lagrangian" or "EHL") Here m is the electron mass, and a, b are the two invariants of the Maxwell field, related to E, B by This Lagrangian holds the information on the QED Nphoton amplitudes in the low-energy limit where all photon energies are small compared to the electron mass, ω i ≪ m.It corresponds to the diagrams shown in Fig. 2.
This formula (called 'AAM formula' in the following) is highly remarkable for various reasons.Despite of its simplicity it is a true all-loop result; the rhs receives contributions from an infinite set of Feynman diagrams of arbitrary loop order, as sketched in fig. 1.For the extraction of the amplitudes from the effective Lagrangian, one expands it in powers of the Maxwell invariants,

Number of external legs
then fixes a helicity assignment and uses spinors helicity techniques [3].

Imaginary part of the effective action
Except for the purely magnetic case where b = 0, the proper-time integral in (1) has poles on the integration contour at e bT = kπ which create an imaginary part.For the purely electric case one gets [4] . We note: • The kth term relates to coherent creation of k pairs in one Compton volume.
• ImL(E) depends on E non-perturbatively (non-analytically), which is consistent with Sauter's [5] interpretation of pair creation as vacuum tunnelling (Fig. 3).building on earlier work of Sauter [18].This result sets a basic scale of a critical field strength and intensity near which we expect to observe such nonperturbative effects:

031.3
As a useful guiding analogy, recall Oppenheimer's computation [19] of the probability of ionization of an atom of binding energy E b in such a uniform electric field: Taking as a representative atomic energy scale the binding energy of hydrogen, E b = me 4 2h 2 ≈ 13.6 eV, we find This result sets a basic scale of field strength and intensity near which we expect to observe such nonperturbative ionization effects in atomic systems: These, indeed, are the familiar scales of atomic ionization experiments.Note that E ionization c differs from E c by a factor of α 3 ∼ 4 × 10 −7 .These simple estimates explain why vacuum pair production has not yet been observed -it is an astonishingly weak effect with conventional lasers [20,21].This is because it is primarily a non-perturbative effect, that depends exponentially on the (inverse) electric field strength, and there is a factor of ∼ 10 7 difference between the critical field scales in the atomic regime and in the vacuum pair production regime.Thus, with standard lasers that can routinely probe ionization, there is no hope to see vacuum pair production.However, recent technological advances in laser science, and also in theoretical refinements of the Heisenberg-Euler computation, suggest that lasers such as those planned for ELI may be able to reach this elusive nonperturbative regime.This has the potential to open up an entirely new domain of experiments, with the prospect of fundamental discoveries and practical applications, as are described in many talks in this conference.

II. THE QED EFFECTIVE ACTION
In quantum field theory, the key object that encodes vacuum polarization corrections to classical Maxwell electrodynamics is the "effective action" Γ[A], which is a functional of the applied classical gauge field A µ (x) [22,23,24].The effective action is the relativistic quantum field theory analogue of the grand potential of statistical physics, in the sense that it contains a wealth of information about the quantum system: here, the nonlinear properties of the quantum vacuum.For example, the polarization tensor Π µν = δ 2 Γ δAµδAν contains the electric permittivity ij and the magnetic permeability µ ij of the quantum vacuum, and is obtained by varying the effective action Γ[A] with respect This formula (called 'AAM formula' in the following) is highly remarkable for various reasons.Despite of its simplicity it is a true all-loop result; the rhs receives contributions from an infinite set of Feynman diagrams of arbitrary loop order, as sketched in fig. 1.

Beyond one loop
The two-loop correction to the EHL due to one internal photon exchange (Fig. 4) has been analyzed [6][7][8], and turned out to contain important information on the Sauter tunnelling picture [9], on-shell versus off-shell renormalization [6,10], and the asymptotic properties of the QED photon S-matrix [11].
It leads to rather intractable two-parameter integrals.However, in the electric case its imaginary part ImL (2) (E) permits a decomposition analogous to Schwinger's (3) [9].For single-pair production, this is now interpreted as a tunnelling process where, in the process of turning real, the electron-positron pair is already interacting at the one-photon exchange level.
Even for the imaginary part no completely explicit formulas are available.However, it simplifies dramatically in the weak-field limit, where it just becomes an απ correction to the one-loop contribution: This suggests that higher loop orders might lead to an exponentiation, and indeed Lebedev and Ritus [9] provided strong support for this hypothesis by showing that, assuming that then the result can be interpreted in the tunnelling picture as the corrections to the Schwinger pair creation rate due to the pair being created with a negative Coulomb interaction energy Moreover, the resulting field-dependent mass-shift δm(E) is identical with the Ritus mass shift, originally derived by Ritus in [13] from the crossed process of one-loop electron propagation in the field (Fig. 5).
Unbeknownst to the authors of [9], for scalar QED the corresponding conjecture had already been established two years earlier by Affleck, Alvarez and Manton [12] using Feynman's worldline path integral formalism and a semi-classical worldline instanton approximation.(E) in the weak-field limit.

031.4
Moreover, the mass appearing in (1.15) is argued to be still the physical renormalized mass, which means that the above figure should strictly speaking include also the mass renormalization counter diagrams which appear in EHL calculations starting from two loops.
The derivation given in [33] is very simple, if formal.Based on a stationary path approximation of Feynman's worldline path integral representation [34] of L scal (E), it actually uses only a one-loop semiclassical trajectory, and arguments that this trajectory remains valid in the presence of virtual photon insertions.This also implies that non-quenched diagrams do not contribute in the limit (1.15), which is why we have shown only the quenched ones in fig. 1.
Although the derivation of (1.15) in [33] cannot be considered rigorous, an independent heuristic derivation of (1.15), as well as extension to the spinor QED case (with the same factor of e απ ) was given by Lebedev and Ritus [31] through the consideration of higher-order corrections to the pair creation energy in the vacuum tunneling picture.At the two-loop level, (1.15) and its spinor QED extension state that 6 Diagrammatically, we note the following features of the exponentiation formula (see Fig. 6): • It Involves diagrams with any numbers of loops and legs.
• Although not shown, also all the counter-diagrams from mass renormalization must contribute.
• It does not include diagrams with more than one fermion loop (those get suppressed in the weak-field limit [12]).

QED in 1+1 dimensions
The exponentiation conjecture has so far been verified only at two loops.A three-loop check is in order, but calculating the three-loop EHL in D = 4 is presently hardly feasible.Motivated by work by Krasnansky [14] on the EHL in various dimensions, in 2010 two of the authors with D.G.C. McKeon started investigating the analogous problem in 2D QED.In [15] we used the worldline instanton method to generalize the exponentiation conjecture to the 2D case, resulting in where κ = m 2 /(2e f ), f 2 = 1 4 F µν F µν , and α = 2e 2 πm 2 is the two-dimensional analogue of the fine-structure constant.Defining the weak-field expansion coefficients in 2D by 031.5 we then used Borel analysis to derive from (3) a formula for the limits of ratios of l -loop to one -loop coefficients: Moreover, we calculated the 2D EHL at one and two loops, where ψ(x) ≡ ψ(x) − ln x + 1 2x , ψ(x) = Γ ′ (x)/Γ (x), and the constant λ 0 comes from an IR cutoff.One finds from ( 6) and ( 7) that c (1) (2) Using properties of the Bernoulli numbers B n it is then easy to verify that lim n→∞ c (2) in accordance with (5).

Three-loop EHL in 2D: Diagrams
At three loops, we face the task of computing the two diagrams shown in Fig. 7 (there are also diagrams involving more than one fermion-loop, including several that involve Gies-Karbstein tadpoles [16], but those can be shown to be subdominant in the asymptotic limit).
The fermion propagators in these diagrams are the exact ones in the constant external field.Thus, although they are depicted as vacuum diagrams, they are equivalent to the full set of ordinary diagrams of the given topology with any number of zero momentum photons attached to them in all possible ways.
Due to the super-renormalizability of 2D QED these diagrams are already UV finite.They suffer from spurious IR -divergences, but those can be removed by going to the traceless gauge ξ = −2 [17].The calculation of diagram A is relatively straightforward, thus we focus on the much more substantial task of computing diagram B and its weak-field expansion coefficients.We will use k, q, and p as the independent variables.The remaining variables are expressed in terms of them as

031.6
With these conventions, the contribution of this diagram is written as 5 where ).Although for a three-loop diagram this is a fairly compact representation, an exact calculation is out of the question, and a straightforward expansion in powers of the external field to get the weak-field expansion coefficients turns out to create huge numerator polynomials.To deal with those, we will now take advantage of the high symmetry of the diagram.

Integration-by-parts algorithm
Introduce the operator d which acts simply on the trigonometric building blocks of the integrand.Integrating by parts with this operator, it is possible to write the integrand of β n , the n-th coefficient of the expansion of I B as a power series in ρ, as a total derivative β n = dθ n .Then, using once more the symmetry of the graph, Those generate the dihedral group D 4 .After a slight generalization to the inclusion of semiinvariants (invariants up to a sign) [18], Theorem 1 can be used to deduce that the numerator polynomials can be rewritten as polynomials in the variable w = w−w ′ + ŵ− w with coefficients that are polynomials in the four D 4 -invariants a, v, j, h, a = w + w ′ + ŵ + w , v = 2(w ŵ + w ′ w) + (w + ŵ)(w ′ + w) , j = a w − 4(w ŵ − w ′ w) , h = a(ww ′ ŵ + ww ′ w + w ŵ w + w ′ ŵ w) + (w ŵ − w ′ w) 2 .
These invariants are moreover chosen such that they are annihilated by d.Thus they are well-adapted to the integration-by-parts algorithm.This rewriting leads to a very significant reduction in the size of the expressions generated by the expansion in the field.

Results
In this way we obtained the first two coefficients of the weak-field expansion analytically,

Outlook
• Writing Feynman graph polynomials in terms of invariant polynomials is a universal option that, to the best of our knowledge, has not previously been used, but we expect that it will be found very useful for multiloop calculations involving diagrams with nontrivial symmetry groups and a large number of propagators.
• In particular, this is the case for the weak-field expansion of the QED effective Lagrangian starting from three loops (in any dimension).

FIG. 1 :
FIG.1: Pair production as the separation of a virtual vacuum dipole pair under the influence of an external electric field.

Figure 3 :
Figure 3: Pair creation by an external field as vacuum tunnelling.

Figure 5 :
Figure 5: Photon-corrected pair-creation vs. electron propagation in the field.

Figure 6 :
Figure 6: Feynman diagrams contributing to the exponentiation hypothesis.
coefficients numerically (the coefficients Γ n are related to the ones introduced in (4) by 4 n c(3)  n = α2 64 Γ n ).For a definite conclusion concerning the exponentiation conjecture this is still insufficient, and the computation of further coefficients is in progress.