One-loop electron mass and QED trace anomaly

Electron mass is considered as a matrix element of the energy–momentum trace in the rest frame. The one-loop diagrams for this matrix element are different from the textbook diagrams for the electron mass renormalization. We clarify connection between the two sets of diagrams and explain analytically and diagrammatically why the results of both calculations coincide.


I. INTRODUCTION
Hadron energy-momentum tensor (EMT), its matrix elements, anomalous trace, form factors and multipole expansion is now a vibrant field of research.EMT form factors describe interaction of particles with weak external gravitational field [1,2].For a long time there was no way to measure form factors of hadron EMT, but the situation changed when it was realized that they are connected to the generalized parton distribution functions which can be measured in deeply virtual Compton scattering and other hard exclusive reactions, see, e.g., [3][4][5][6][7][8].
A new insight into the EMT properties could arise from consideration of EMT in theories which allow perturbative treatment.While perturbative approach is clearly impossible in QCD, one can consider a simpler gauge theory, namely QED, and hope to acquire some experience which would be useful for the hadronic world.One-loop QED contributions to the EMT form factors, matrix elements and trace were calculated for a free electron and electron in the Coulomb field in a number of old and recent papers [17][18][19][20][21][22][23][24][25][26][27][28].
One-loop electron mass renormalization in the mass-shell renormalization scheme is a textbook problem discussed in every introductory quantum field theory textbook, see, e.g., [29].Consideration of EMT suggests another perspective on this classical problem.One can calculate electron mass as a matrix element of the EMT trace.The diagrams describing this matrix element do not coincide with the well known diagrams for the one-loop corrections to the electron mass.We will calculate electron mass with the help of both sets of different contributions and explain why they produce coinciding results.

II. MATRIX ELEMENTS OF EMT AND MASS OF PARTICLES
General formulae for EMT T µν (x) follow from its definition as a conserved two-index symmetric tensor.Due to translational invariance where |p is a particle eigenstate with momentum p and states here are normalized rela- The Hamiltonian is a three-dimensional integral, H = d 3 xT 00 (x), and on the one hand and on the other hand (see Eq. ( 1)) Hence, and due to Lorentz invariance In the rest frame and with the nonrelativistic normalization of states These relations hold both elementary particles and for bound states, and are obviously valid in any relativistic field theory.Below we will consider the first of these equations for an electron in QED.
Symmetric EMT tensor is conserved in a translationally invariant relativistic field theory and it is not renormalized as any conserved operator EMT trace in gauge theories acquires an anomalous contribution [30][31][32] and has the form where β(e)/2e = α/6π, γ m (e) = 3α/2π.
The left hand side in Eq. ( 7) is renorminvariant and then the sum of the operators on the right hand side (RHS) is also renorminvariant.There are subtleties with separation of the terms on the right hand side in a sum of renorminvariant operators beyond one loop, see [15,[33][34][35][36][37][38].
We are going to consider matrix element of the anomalous trace in Eq. ( 7) for an electron at rest in the one-loop approximation.We will be working in the renormalized perturbation theory and use the mass-shell renormalization scheme.Then, according to Eq. ( 6), this matrix element should be equal to the physical electron mass m and describe one-loop mass renormalization.At the same time the diagrams which contribute to this matrix element do not coincide with the well known mass renormalization diagrams.Our goal is to clarify from the diagrammatic and analytic perspectives, why two different diagrammatic descriptions lead to the identical results1 .

III. ONE-LOOP MASS RENORMALIZATION AND EMT ANOMALOUS TRACE FOR A FREE ELECTRON
Let us recall one-loop electron mass renormalization in the mass-shell scheme with dimensional regularization.We collected the well known relevant formulae in the Appendix.In the mass-shell renormalization scheme the counterterm δm (2) kills the ultraviolet divergence in the regularized but not renormalized self-energy diagram Σ(p) and preserves the physical where δm (2) = Σ(m).This expression is illustrated in Fig. 1.The factor i before the selfenergy diagram in Fig. 1 is included in the standard diagrammatic definition of Σ(p), see e.g., [29].
Next we turn to the matrix element of the EMT trace in Eq. ( 6) for the electron.The leading contribution to the matrix element of the term (β(e 0 )/2e 0 )F 2 0 in Eq. ( 7) is of order α 2 and we ignore it.It is sufficient to calculate the one-loop matrix element In the one-loop approximation = m + mδZ 2 − δm (2) where Γ m (m) is the one-loop diagram for the scalar vertex m ψψ, see Fig. 2.
2. One-loop matrix element of the EMT trace.
All terms on the RHS in Eq. ( 10) and in Fig. 2, except Γ m (m), are known from the one-loop mass-shell renormalization scheme (see Eq. (A7) and Eq.(A8)) and only Γ m (m) requires calculation.After an easy one-loop calculation we obtain where λ is the infrared photon mass and γ is the Euler constant.
Next we use Γ m (m), the renormalization constants in Eq. (A7) and Eq.(A8), and γ m = 3α/2π to calculate the sum in Eq. ( 10) Thus we confirmed that the matrix element T of the anomalous EMT trace in the one-loop approximation is equal the physical electron mass, as it should be.Comparing Eq. ( 8) and Eq. ( 12) (and the respective Figs. 1 and 2) we see that At this stage it is unclear why the different sets of diagrams in Fig. 1 and Fig. 2 produce coinciding results.To figure out a deeper reason why this happens we expand the unrenormalized electron self-energy Σ( / p) in the Taylor series near the physical mass Differentiating with respect to m we obtain at Notice that (see Fig. 3) which holds due to the identity Then Eq. ( 15) can be written in the form and Eq. ( 13) turns into (δm (2) = Σ( / p = m)) We calculate the derivative on the RHS using the explicit expression for δm (2) in Eq. (A7) and obtain (see Fig. 4) where at the last step we used the definition of the electron mass anomalous dimension.
Notice that this relationship holds due to a specific functional form of the mass counterterm δm (2) = mf (m/µ) with some function f .Thus we proved by direct calculation that Eq. ( 19) holds and the expressions in Eq. (8) and Eq. ( 12) (and the respective sets of diagrams in Fig. 1 and Fig. 2) coincide.

IV. CONCLUSIONS
We have shown that the standard mass renormalization in Fig. 1 and the sum of the diagrams for the matrix element of the EMT trace in Fig. 2 coincide.This happens due to two important relationships.First, the one-loop diagram for a scalar vertex is equal to the logarithmic derivative of the self-energy diagram, see Eq. ( 16) and Fig. 3. Second, the mass renormalization counterterm (self-energy at / p = m) is equal to its own logarithmic derivative plus the product of mass and its anomalous dimension, see Eq. (19) and Fig. 4.
The calculations above are made in the one-loop approximation, but we expect that they, including the functional relationship mentioned after Eq. ( 20), can be generalized to any number of loops.Really, connection between an arbitrary diagram and its logarithmic derivative with respect to the fermion mass, and the connection between such derivative and the fermion mass anomalous dimension do not depend on the number of loops, and these are the only essential steps in the derivation above.
Appendix A: Standard one-loop electron mass renormalization Some well known results are collected below.We use dimensional regularization and mass-shell renormalization.The QED Lagrangian in this scheme is where and We define δm = m − m 0 = m − mZ m Z −1 2 .In the one-loop approximation δm