Nonrelativistic Electron in the Coulomb Field
In one-electron atoms, the energy levels can be understood through the solutions of the Schrödinger equation, which describes the behavior of an electron in the electric field of an infinitely heavy Coulomb center with a charge of Z, equivalent to the proton charge.
In quantum mechanics, the principal quantum number \( n \) (where \( n = 1, 2, 3, \ldots \)) defines the energy levels of an electron in an atom Each state is further characterized by the orbital angular momentum \( l \) (ranging from 0 to \( n-1 \)) and its projection \( m \) (which can take values from \( -l \) to \( +l \)) In the nonrelativistic Coulomb problem, states with different orbital angular momentum but the same principal quantum number \( n \) possess identical energy levels, resulting in \( n \)-fold degeneracy Additionally, due to the spherical symmetry of the Coulomb field, the energy levels remain unaffected by the projection of orbital angular momentum along any axis, leading to an additional \( 2l + 1 \)-fold degeneracy for each energy level defined by \( l \).
1 We are using the system of units where ¯h=c= 1.
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2 1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems
Straightforward calculation of the characteristic values of the velocity, Coulomb potential and kinetic energy in the stationary states gives n|v 2 |n n p 2 m 2 n
The fine structure constant α indicates that a one-electron atom behaves as a loosely bound nonrelativistic system, allowing relativistic effects to be treated as perturbations This system is defined by three characteristic scales.
2 We are interested only in low-Z atoms High-Z atoms cannot be treated as non- relativistic systems, since an expansion inZαis problematic.
Dirac Electron in the Coulomb Field
In an atom, the smallest scale is influenced by binding energy, approximated as ∼m(Zα)² The subsequent scale is governed by the characteristic momenta of electrons, represented as ∼mZα Finally, the last scale is on the order of the electron mass, denoted as m.
In nonrelativistic quantum mechanics, a more accurate representation of the hydrogen spectrum can be obtained by considering the finite mass of the Coulomb center This approach necessitates the use of the reduced mass in place of the electron mass due to the bound system's nonrelativistic nature The finite mass of the nucleus introduces the most significant energy scale in the problem of the bound system, which is the mass of the heavy particle.
1.2 Dirac Electron in the Coulomb Field
The relativistic dependence of the energy of a free classical particle on its momentum is described by the relativistic square root p 2 +m 2 ≈m+ p 2
The kinetic energy operator in the Schrödinger equation represents the quadratic term in the nonrelativistic expansion, indicating that the equation provides only the primary nonrelativistic approximation of hydrogen's energy levels.
The classical nonrelativistic expansion involves the term p²/m², which, for loosely bound electrons, corresponds to an expansion in (Zα)² Consequently, relativistic corrections arise from this expansion over even powers of Zα As demonstrated by the explicit energy level expressions in the Coulomb field, the parameter Zα also characterizes the binding energy Therefore, Zα is frequently referred to as the binding parameter, and the relativistic corrections are commonly known as binding corrections.
The series expansion for relativistic corrections in the bound state problem directly involves the binding parameter Zα, contrasting with the scattering problem in quantum electrodynamics (QED), where the expansion parameter includes an additional factor of π in the denominator, typically expressed as α/π This distinction is a hallmark of the Coulomb problem In combined expansions over α and Zα, the expansion over α at a fixed power of Zα also adopts the α/π form, similar to scattering scenarios The successive terms in the series over Zα can be referred to as relativistic corrections, while those in the α/π expansion represent loop or radiative corrections Additionally, when calculating relativistic corrections for the bound electron, it is essential to consider contributions from its spin one half, although this consideration does not alter the fundamental nature of the relativistic effects involved.
4 1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems
Relativistic corrections are expressed through the expansion in even powers of Zα, similar to the classical relativistic square root expansion In this context, only the coefficients of the expansion are affected by the presence of spin The Dirac equation with a Coulomb source provides a comprehensive description of all relativistic corrections to energy levels These corrections can be readily derived from the exact solution of the Dirac equation in an external Coulomb field, as referenced in sources [1, 2].
+ã ã ã , (1.5) andj = 1/2,3/2, , n−1/2 is the total angular momentum of the state.
In the Dirac spectrum, energy levels with the same principal quantum number (n) but different total angular momentum (j) are split into multiple components of fine structure, contrasting with the nonrelativistic Schrödinger spectrum where levels with the same n are degenerate However, some degeneracy persists in the Dirac equation spectrum, as energy levels corresponding to the same n and j but different l = j±1/2 remain doubly degenerate This degeneracy is lifted by corrections related to the finite size of the Coulomb source, recoil contributions, and significant quantum electrodynamic (QED) loop contributions, resulting in energy shifts known as Lamb shifts Notably, quantum mechanical effects alone, such as recoil and finite nuclear size, do not account for the scale of the experimentally observed Lamb shift, underscoring its nature as a fundamentally QED effect.
A simple enhancement to the Dirac formula for energy levels can be made by acknowledging that electron motion in a Coulomb field is primarily nonrelativistic Consequently, all contributions to the binding energy should incorporate the reduced mass of the electron-nucleus system instead of just the electron mass We will examine the expression that includes the reduced mass factor.
The equation E nj = m + m r [f(n, j) - 1] serves as a more accurate foundation for calculating corrections to electron energy levels, compared to the simpler expression in (1.4) To ensure a reliable basis for subsequent calculations, the Dirac spectrum is considered with the reduced mass.
Bethe-Salpeter Equation and the Effective Dirac Equation
dependence in (1.6) should be itself derived from QED (see Sect 3.1 below), and not simply postulated on physical grounds as is done here.
1.3 Bethe-Salpeter Equation and the Effective Dirac Equation
Quantum field theory offers a clear method for determining the energy levels of composite systems, which are identified by the positions of the poles in the corresponding Green functions This concept was initially formulated through the Bethe-Salpeter (BS) equation, specifically for the two-particle Green function.
[3] G=S 0+S 0 K BS G , (1.7) whereS 0is a free two-particle Green function, the kernelK BS is a sum of all two-particle irreducible diagrams in Fig 1.3, andG is the total two-particle Green function.
The field-theoretical Bethe-Salpeter (BS) equation may initially appear unrelated to the quantum mechanical Schrödinger and Dirac equations; however, by selecting specific interaction kernels and applying natural approximations, it can be shown that the BS eigenvalue equation simplifies to the Schrödinger or Dirac equations for a light particle influenced by a heavy Coulomb center The fundamentals of the BS equation are well-documented in various textbooks, and significant results have been achieved within the BS framework.
Calculations beyond the leading order in the original Bethe-Salpeter (BS) framework are often complex and opaque This complexity arises from the BS wave function's reliance on unphysical relative energy or relative time, as well as the lack of an exact solution.
Fig 1.3.Kernel of the Bethe-Salpeter equation
Theoretical approaches to the energy levels of loosely bound systems face significant challenges, particularly in the zero-order approximation and the limitations of the ladder approximation in relation to the Dirac equation as the mass of heavy particles approaches infinity These challenges arise not only from the nonpotential nature of the bound state problem in quantum field theory but also from the unphysical classification of diagrams based on the concept of two-body reducibility Historically, there has been a notable tendency for cancellation between contributions from ladder graphs and those with crossed photons; however, the original Bethe-Salpeter (BS) framework treats these graphs in fundamentally different manners Consequently, it is essential to modify the BS equation to ensure a more symmetrical treatment of crossed and ladder graphs while addressing other drawbacks of the original formulation, all while maintaining its rigorous field-theoretical foundations.
The BS equation is highly adaptable, allowing modifications to both the zero-order propagation function and the leading order kernel, provided that these changes are consistently integrated into the rules for constructing higher order approximations, aligned with the two-particle Green function Since its inception, various versions of the BS equation have emerged, focusing on restructuring it into a three-dimensional format This approach aims to yield a solvable and physically intuitive leading order approximation, resembling the Schrödinger or Dirac equations, while also facilitating a clear and systematic selection of kernels for calculating corrections of any desired order.
This article discusses the Effective Dirac Equation (EDE), a modification derived in several studies, which proves to be more advantageous than the original BS equation for various applications The EDE is particularly useful for analyzing loosely bound two-particle systems with different masses, where the heavier particle remains close to its mass shell This context allows for the formulation of a Dirac equation for the lighter particle within an external Coulomb field, serving as a solid foundation for perturbation theory expansion Additionally, the article presents a convenient choice for the free two-particle propagator, expressed as the product of the heavy particle mass shell projector and the free electron propagator.
E/−p/−m(2π) 4 δ (4) (p−l), (1.8) wherep à andl à are the momenta of the incoming and outgoing heavy particle,
The momentum of the incoming electron is represented as E à −p à, where E = (E, 0) based on the selected reference frame The γ-matrices relevant to both light and heavy particles function solely on the indices of their respective particles.
1.3 Bethe-Salpeter Equation and the Effective Dirac Equation 7
The free propagator in equation (1.8) establishes the foundational components and the structure of a two-body equation that corresponds to the BS equation, with the regular perturbation theory formulas derived in references [9, 10].
In order to derive these formulae let us first write the BS equation in (1.7) in an explicit form
S 0 (p, k, E) = i p /−M i E/−/l−m(2π) 4 δ (4) (p−l) (1.10) The amputated two-particle Green functionG T satisfies the equation
A new kernel corresponding to the free two-particle propagator in (1.8) may be defined via this amputated two-particle Green function
Comparing (1.11) and (1.12) one easily obtains the diagrammatic series for the new kernelK (see Fig 1.4)
The new bound state equation is constructed for the two-particle Green function defined by the relationship
The two-particle Green functionG has the same poles as the initial Green functionG and satisfies the BS-like equation
Fig 1.4 Series for the kernal of the effective Dirac equation
8 1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems
This last equation is completely equivalent to the original BS equation, and may be easily written in a three-dimensional form
(1.17) where all four-momenta are on the mass shell p 2 = l 2 = q 2 = M 2 , E q q 2 +M 2 , and the three-dimensional two-particle Green function G is de- fined as follows
Taking the residue at the bound state pole with energy E n we obtain a ho- mogeneous equation
The wave function in equation (1.19) adheres to a free Dirac equation concerning the heavy particle indices, attributed to the influence of the heavy particle mass shell projector on the right side.
Then one can extract a free heavy particle spinor from the wave function in (1.19) φ( p, E n ) = 2E n U( p )ψ( p, E n ), (1.21) where
Finally, the eight-component wave function ψ( p, E n ) (four ordinary electron spinor indices, and two extra indices corresponding to the two-component
1.3 Bethe-Salpeter Equation and the Effective Dirac Equation 9
Fig 1.5.Effective Dirac equation spinor of the heavy particle) satisfies the effective Dirac equation (see Fig 1.5)
, (1.24) k= (E n −p 0 ,−p ) is the electron momentum, and the crosses on the heavy line in Fig 1.5 mean that the heavy particle is on its mass shell.
The inhomogeneous equation (1.17) also fixes the normalization of the wave function.
While the total kernel in equation (1.23) is clearly defined, we have the flexibility to select the zero-order kernel \( K_0 \) to achieve a manageable lowest order approximation It is straightforward to derive a regular perturbation theory series that accounts for the corrections to the zero-order approximation, reflecting the difference between the chosen zero-order kernel \( K_0 \) and the exact kernel \( K_0 + \delta K \).
+ã ã ã , (1.25) where the summation of intermediate states goes with the weight d 3 p/
[(2π) 3 2E p ] and is realized with the help of the subtracted free Green func- tion of the EDE with the kernelK 0
In the context of the Dirac equation, conjugation is defined in the Dirac sense, with the notation δK (E n 0) representing the derivative of K with respect to energy at a specific energy level The Energy-Dependent Equation (EDE) differs from the standard Dirac equation primarily due to the energy-dependent interaction kernels Consequently, the perturbation theory series in this framework includes energy derivatives of the interaction kernels, which play a vital role in eliminating ultraviolet divergences from the energy eigenvalue expressions.
A judicious choice of the zero-order kernel (sum of the Coulomb and Breit potentials, for more detail see, e.g, [6, 7, 10]) generates a solvable unperturbed
10 1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems
Fig 1.6.Effective Dirac equation in the external Coulomb field
In the context of the external Coulomb field, the eigenfunctions can be precisely represented by the Dirac-Coulomb wave functions However, for practical applications, it is often sufficient to use an approximation involving the product of the Schrödinger-Coulomb wave functions with the reduced mass and free electron spinors, which are dependent on the electron mass These approximated functions facilitate the calculation of high-order corrections Although some derivation steps may be omitted in the following sections, it is important to note that most calculations are based on these unperturbed wave functions.
Nonrelativistic Quantum Electrodynamics
Weakly bound states are inherently nonrelativistic, characterized by velocities approximately equal to Zα, where α is the fine structure constant These states can be described using quantum mechanics without requiring quantum field theory in leading approximations However, nonrelativistic quantum mechanical approaches struggle to accurately calculate higher order corrections when recoil and many-particle effects are significant In contrast, the Bethe-Salpeter equation offers a comprehensive quantum field theory framework for analyzing both weakly and strongly bound states Despite its generality, the Bethe-Salpeter formalism complicates the separation of nonrelativistic dynamics for weakly bound states and makes the systematic extraction of higher order corrections increasingly challenging.
Nonrelativistic quantum electrodynamics (NRQED) aims to merge the straightforwardness of quantum mechanics with the rigor of field theory By expressing relativistic quantum electrodynamics as a nonrelativistic expansion, NRQED utilizes a Lagrangian that incorporates vertices with various powers of fields This approach is particularly beneficial for analyzing nonrelativistic processes, such as bound states and threshold phenomena, where the primary dynamics are nonrelativistic, potentially simplifying calculations significantly.
3 Strictly speaking the external field in this equation is not exactly Coulomb but also includes a transverse contribution.
In our analysis, we can organize expansions in terms of the fine-structure constant α and the nonrelativistic velocity v, which is determined by kinematics in scattering processes For nonrelativistic bound states, this velocity relates to the coupling constant, serving as a useful small parameter even when perturbation theory is not applicable The approach involves introducing a finite cutoff Λ, typically chosen as Λ∼m for bound states, allowing us to utilize two small parameters: α and p/Λ∼p/m∼v We then compare coefficients for nonrelativistic vertices within the framework of relativistic quantum electrodynamics (QED) and nonrelativistic QED (NRQED), ensuring that comparisons are made at a fixed velocity v and to arbitrary orders in α, while avoiding ultraviolet convergence issues due to the finite cutoff This methodology has been widely implemented in the literature, both in explicit cutoff schemes and dimensional regularization, with a focus on high-order corrections to energy levels using effective operators of NRQED These operators often have clear and intuitive origins, making them more accessible than lengthy formal derivations.
1 J D Bjorken and S D Drell, Relativistic Quantum Mechanics, McGraw-Hill Book Co., NY, 1964.
2 V B Berestetskii, E M Lifshitz, and L P Pitaevskii, Quantum electrodynam- ics, 2nd Edition, Pergamon Press, Oxford, 1982.
3 E E Salpeter and H A Bethe, Phys Rev 84, 1232 (1951).
4 C Itzykson and J.-B Zuber, Quantum Field Theory, McGraw-Hill Book Co.,
5 F Gross, Relativistic Quantum Mechanics and Field Theory, Wiley, NY, 1993.
6 H Grotch and D R Yennie, Zeitsch Phys.202, 425 (1967).
7 H Grotch and D R Yennie, Rev Mod Phys.41, 350 (1969).
9 L S Dulyan and R N Faustov, Teor Mat Fiz 22, 314 (1975) [Theor Math.
11 W E Caswell and G P Lepage, Phys Lett B167, 437 (1986).
12 1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems
12 T Kinoshita and M Nio, Phys Rev D53, 4909 (1996).
13 G P Lepage, preprint nucl-th/9706029, February 1997.
15 A Pineda and J Soto, Phys Rev D 59, 016005 (1999).
16 A H Hoang, preprint hep-ph/0204299, April 2002.
General Features of the Hydrogen Energy Levels
Classification of Corrections
The zero-order effective Dirac equation with a Coulomb source offers an approximate representation of loosely bound states in quantum electrodynamics (QED) However, the spectrum derived from this Dirac equation can be utilized as a valuable foundation for achieving more accurate results in the study of these states.
The Dirac equation with a Coulomb source overlooks the magnetic moment of heavy nuclei, resulting in the absence of hyperfine splitting (HFS) in its energy level spectrum However, the magnetic interaction between the nucleus and the electron can be effectively described using nonrelativistic quantum mechanics, with Fermi having previously calculated the leading contribution to hyperfine splitting.
In quantum mechanics, corrections to Dirac energy levels and hyperfine splitting require field-theoretical methods, as they do not emerge from standard treatments with potentials Electrodynamic corrections can be expressed as a power series expansion involving three small parameters: α, Zα, and m/M, which characterize the bound state properties Additionally, nonelectromagnetic corrections from strong and weak interactions introduce further small parameters, such as the ratio of nuclear radius to Bohr radius and the Fermi constant It is important to recognize that the coefficients in the energy level power series may vary slowly and can be logarithmic functions of these parameters.
Each of the small parameters above plays an important and unique role.
In order to organize further discussion of different contributions to the energy levels it is convenient to classify corrections in accordance with the small parameters on which they depend.
Corrections which depend only on the parameter Zαwill be called rela- tivisticor binding corrections Higher powers of Zαarise due to deviation of
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The hydrogen energy levels can be understood through a nonrelativistic framework, which allows for a relativistic expansion This approach incorporates all relevant contributions within the spectrum described by the effective Dirac equation in the presence of an external Coulomb field.
Radiative corrections, which are contributions to energy dependent solely on the small parameters α and Zα, arise from quantum electrodynamics (QED) loops and possess a quantum field theory nature These corrections are independent of the recoil factor m/M, allowing for calculations within QED for a bound electron in an external field While these calculations involve complexities related to quantized fields, they can overlook the two-particle nature of the bound state and the challenges associated with describing bound states in relativistic quantum field theory.
Corrections related to the mass ratio \( m/M \) of light and heavy particles indicate a deviation from the theoretical model that assumes an infinitely heavy nucleus These energy level corrections, which depend on both \( m/M \) and \( Z\alpha \), are known as recoil corrections They account for contributions to energy levels that cannot be addressed using the reduced mass factor, highlighting that the system is a genuine two-body problem rather than a simplified one-body problem.
Recoil corrections in Zα, specifically of order (Zα)⁴ (m/M)ⁿ, can be effectively addressed using the Dirac equation in external fields due to their one-photon exchange origin However, higher-order recoil terms that represent the complex two-body nature of bound states cannot be handled in the same manner These contributions arise from the Bethe-Salpeter kernels, necessitating at least two-photon exchanges and a comprehensive framework of relativistic quantum field theory (QFT) for accurate calculations Fortunately, the calculation of these recoil corrections is simplified by the absence of ultraviolet divergences associated with purely radiative loops.
Radiative-recoil corrections are essential expansion terms in energy level expressions, influenced by parameters such as α, m/M, and Zα Calculating these corrections necessitates the use of advanced techniques and methodologies.
QED, since we have to account both for the purely radiative loops and for the relativistic two-body nature of the bound states.
The final category of corrections includes nonelectromagnetic effects arising from weak and strong interactions, with the most significant correction related to the finite size of the nucleus It is important to note that the Dirac energy spectrum does not account for hyperfine structure, radiative, recoil, radiative-recoil, and nonelectromagnetic corrections This article will focus on the calculations of these missing corrections.
Physical Origin of the Lamb Shift
2.2 Physical Origin of the Lamb Shift
Quantum Electrodynamics (QED) describes how an electron continuously emits and absorbs virtual photons, leading to the distribution of its electric charge over a finite volume rather than being pointlike This phenomenon is illustrated in the leading order diagram (Fig 2.1) and can be quantified by the equation \( 1/r^2 = -6 dF_1(-k^2) dk^2 |_{k^2=0} \approx \frac{2\alpha}{\pi m^2} \ln \rho \approx \frac{2\alpha}{\pi m^2} \ln(Z\alpha) - 2 \).
To estimate the electron radius, we consider the electron's slight off-mass-shell condition in its bound state This results in an infrared divergence in the electron charge radius, which is mitigated by its virtuality, expressed as ρ = (m² - p²)/m This virtuality is approximately equal to the nonrelativistic binding energy, given by ρ ≈ m(Zα)².
Fig 2.1.Leading order contribution to the electron radius
The finite radius of the electron generates a correction to the Coulomb potential (see, e.g., [2]) δV =1
3 Zαr 2 δ( r ), (2.3) whereV =−Zα/r is the Coulomb potential.
The matrix element of the perturbation provides the necessary correction to the energy levels, revealing that the finite size of the electron, due to QED radiative corrections, results in a shift of hydrogen energy levels This perturbation, which is significant only at the source of the Coulomb potential, affects energy levels with varying orbital angular momenta differently, leading to a splitting of levels that possess the same total angular momentum but different orbital momenta Consequently, this splitting removes the degeneracy in the spectrum of the Dirac equation in a Coulomb field, where energy levels are solely dependent on the principal quantum number \( n \) and total angular momentum \( j \).
1 The numerical factor in (2.2) arises due to the common relation between the expansion of the form factor and the mean square root radius
16 2 General Features of the Hydrogen Energy Levels
It is very easy to estimate this splitting (shift of theS level energy)
This result should be compared with the experimental number of about
The energy shift at 1040 MHz provides a satisfactory crude estimate, highlighting two key qualitative features Firstly, the energy shift's sign can be deduced without calculations, as the electron's finite radius causes its charge to be more dispersed around the Coulomb source, resulting in weaker binding and a higher energy level compared to a pointlike electron Secondly, although nonlogarithmic contributions are absent in this basic calculation, their impact is minimal, with the logarithmic term being the primary contributor to the Lamb shift This characteristic arises from the potential infrared divergence of the contribution, which is mitigated by the relatively small binding energy compared to the electron mass.
As we will see below, whenever a correction is logarithmically enhanced, the respective logarithm gives a significant part of the correction, as is the case above.
The Lamb shift's leading order contributions include a significant correction associated with polarization insertion in the photon propagator This correction results in an adjustment to the Coulomb potential, represented as δV = −Π(−k²) k⁴ |k²=0.
15 α(Zα) m 2 δ( r ), (2.5) and the respective correction to theS-level energy is equal to
Once again the sign of this correction is evident in advance The polariza- tion correction may be thought of as a correction to the electric charge of
Fig 2.2.Leading order polarization insertion
Natural Magnitudes of Corrections to the Lamb Shift
The interaction between the electron and proton at a finite distance leads to the nucleon effect, where the electron, having penetrated the polarization cloud, perceives an enhanced charge and experiences a stronger binding force, resulting in a lower energy level This phenomenon significantly contributed to the Lamb shift, providing crucial experimental evidence for modern quantum electrodynamics by confirming the existence of closed electron loops While the vacuum polarization contribution is numerically less significant than the quantum corrections related to electron spreading, the overall shift in energy levels remains positive.
2.3 Natural Magnitudes of Corrections to the Lamb Shift
The primary contribution to the Lamb shift arises from a radiative correction that includes a logarithmic enhancement factor, which is crucial for understanding the significance of higher-order corrections This logarithmic enhancement complicates the ability to estimate the magnitude of these corrections by merely comparing them to the leading order contribution Instead, it is more effective to identify a skeleton factor from the leading order contributions, which can then serve as a normalization factor for further analysis.
4m(Zα) 4 n 3 ×radiative correction, (2.7) where the radiative correction is either the slope of the Dirac form factor, roughly speaking equal tom 2 dF 1 (−k 2 )/dk 2 |k 2 =0 =α/(3π) ln(Zα) −2 , or the polarization correctionm 2 Π(−k 2 )/k 4 | k 2 =0 =α/(15π).
The scale setting factor for qualitative estimates of high-order corrections to the Lamb shift is identified as 4m(Zα)⁴/n³, highlighting a significant dependence on the principal quantum number, 1/n³, which is derived from the wave function's behavior at the origin, |ψ(0)|² ∼ 1/n³ Corrections that occur at small distances or high virtual momenta exhibit this state-independent characteristic Additionally, the coefficients preceding the leading powers of the low-energy logarithms, ln(Zα)², are also state-independent, as these logarithms arise from integrating over a broad range of intermediate momenta, from m(Zα)² to m, with the factor before the logarithm being determined accordingly.
2 We remind the reader that according to the common renormalization procedure the electric charge is defined as a charge observed at a very large distance.
The general features of hydrogen energy levels are influenced by the high momentum region of the integration When estimating higher-order corrections to the Lamb shift, it's important to note that, unlike radiative corrections to scattering amplitudes, the bound state problem does not include an additional π factor in the denominator with factors of Zα This characteristic of the Coulomb problem underscores the necessity of maintaining Z in analytic expressions.
Z = 1), since in this way one may easily separate powers ofZαnot accompa- nied byπ from powers ofαwhich always enter formulae in the combination α/π.
2 J D Bjorken and S D Drell, Relativistic Quantum Mechanics, McGraw-HillBook Co., NY, 1964.
This article explores the corrections to the fundamental Dirac energy levels that emerge from the external field approximation, focusing on the leading relativistic corrections that incorporate precise mass dependence and radiative effects.
Leading Relativistic Corrections with Exact Mass Dependence 19
In a loosely bound two-particle system, the binding energy primarily relies on the nonrelativistic reduced mass, with the main contribution being proportional to (Zα)², rather than the individual masses of the particles The non-relativistic Schrödinger equation fails to account for relativistic corrections of order (Zα)⁴, which are essential for accurately describing the fine structure of the hydrogen spectrum While the relativistic Dirac equation provides insights into the fine structure for an infinitely heavy Coulomb center, it does not clarify the mass dependence of these corrections for nuclei of finite mass Consequently, it is anticipated that relativistic corrections in systems with finite masses will exhibit a more complex relationship with the constituent masses, rather than depending solely on the reduced mass.
The effective Hamiltonian framework provides a solution to the issue of proper mass dependence in the relativistic corrections of order (Zα)⁴ In the center of mass system, the nonrelativistic Hamiltonian for a two-particle system with Coulomb interaction is defined accordingly.
In a nonrelativistic loosely bound system expansion over (Zα) 2 corresponds to the nonrelativistic expansion over v 2 /c 2 Hence, we need an effective
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DOI 10.1007/3-540-45270-2 3 c Springer-Verlag Berlin Heidelberg 2007
The Hamiltonian that includes first-order corrections of order \( \frac{1}{c^2} \) is essential for accurately describing the relative order corrections \( (Z\alpha)^2 \) to nonrelativistic energy levels This concept was initially introduced by Breit, who noted that all first-order corrections to the nonrelativistic two-particle Hamiltonian can be derived from the sum of the free relativistic Hamiltonians of each particle, along with the effects of relativistic one-photon exchange The intuition behind this is clear, as additional exchange photons contribute at least one extra factor of \( Z\alpha \), thereby affecting the binding energy at this order.
An explicit expression for the Breit potential was derived in [2] from the one-photon exchange amplitude with the help of the Foldy-Wouthuysen trans- formation 1
A simplified derivation of the Breit interaction potential may be found in many textbooks (see, e.g., [3]).
All contributions to the energy levels up to order (Zα) 4 may be calculated from the total Breit Hamiltonian
H Br =H 0+V I , (3.3) where the interaction potential is the sum of the instantaneous Coulomb and Breit potentials in Fig 3.1.
Fig 3.1.Sum of the Coulomb and Breit kernels
The corrections of order (Zα)⁴ represent the first-order matrix elements of the Breit interaction between the Coulomb-Schrödinger eigenfunctions of the Coulomb Hamiltonian H₀ The mass dependence of the Breit interaction and its matrix elements is precisely known, allowing for an exact understanding of how these contributions affect energy levels of order (Zα)⁴, extending beyond the reduced mass This critical information regarding the mass dependence was established many years ago.
1 We do not consider hyperfine structure now and thus omit in (3.2) all terms in the Breit potential which depend on the spin of the heavy particle.
3.1 Leading Relativistic Corrections with Exact Mass Dependence 21
The introduction of the final term in equation (3.4) eliminates the characteristic degeneracy in the Dirac spectrum for levels with identical j and l values, specifically j ± 1/2 Consequently, the energy level expression in (3.4) indicates a significant contribution to the classical Lamb shift, denoted as E(2S 1).
The Lamb shift in hydrogen is primarily influenced by quantum electrodynamics (QED) radiative corrections, which have a significantly larger impact than the negligible effect caused by the small electron-proton mass ratio.
The Breit Hamiltonian presented in (3.2) excludes terms related to the spin variables of the heavy particle, leading to energy level corrections in (3.4) that are independent of the spin orientations of both heavy and light particles, meaning hyperfine splitting is not described Most contributions in (3.4) are unaffected by the spins' mutual orientation or the heavy particle's spin magnitude, with the exception of the small Darwin-Foldy contribution, which is relevant only for spin one-half nuclei and should be disregarded for spinless or spin one nuclei This contribution, which naturally integrates with the nuclear size correction, will be further discussed in Subsect 6.1.2.
The effective Dirac equation in the presence of an external Coulomb field was initially derived in reference [4], with subsequent confirmations in sources [5, 6] This result was later rederived in [7], where it was reformulated for clarity.
This equation shares contributions of order (Zα)⁴ similar to those in (3.4), but it also includes nonrecoil and recoil corrections of order (Zα)⁶ and above The nonrecoil contributions are definitively accurate, as the Dirac energy spectrum serves as the correct limit for the spectrum of a two-particle system in the nonrecoil scenario (m/M = 0) Furthermore, we will demonstrate that the first-order mass ratio contributions in (3.5) accurately reflect higher-order recoil corrections in (Zα) generated by Coulomb and Breit exchange photons Additionally, there are first-order mass ratio recoil contributions of order (Zα)⁶.
2 We remind the reader that the external field in this equation also contains a transverse contribution.
The calculation of the 22 3 External Field Approximation will be presented below It is important to note that recoil corrections of order (Zα) 8 have not been computed, and currently, the mass dependence of these terms remains entirely unknown.
Recoil corrections associated with odd powers of Zα are absent in the given equation, as all one-photon exchange corrections rely on even powers of Zα Therefore, to accurately compute recoil corrections of order (Zα) 5, it is essential to include the significant contributions from the box diagram A detailed discussion of these corrections will be provided in Section 4.1.
The Lamb shift refers to a specific phenomenon in quantum electrodynamics where energy levels of certain atomic states are altered due to interactions with virtual photons Initially, researchers focused on measuring the classical Lamb splitting, specifically the energy difference between the 2S and 2P states of hydrogen This shift highlights the significance of quantum effects in atomic structure, providing crucial insights into the behavior of electrons in atoms.
The Lamb shift refers to the energy level splitting observed between degenerate states in the Coulomb field, as predicted by the naive Dirac equation This phenomenon is experimentally measurable and is independent of theoretical interpretations Recent high-precision experiments have provided significant data for the nondegenerate 1S energy level, leading to complexities in defining the Lamb shift clearly It is most appropriately described as the cumulative effect of all contributions that remove the double degeneracy in the Dirac-Coulomb spectrum, specifically concerning the quantum numbers l=j±1/2.
The Lamb shift is conventionally defined as the total contributions to energy levels that exceed the initial three terms in equation (3.5), while excluding hyperfine splitting effects.
We will adopt this definition below.
Radiative Corrections of Order α n (Zα) 4 m
Leading Contribution to the Lamb Shift
3.2.1.1 Radiative Insertions in the Electron Line and the Dirac Form Factor Contribution
The Lamb shift, initially estimated in a nonrelativistic framework by Bethe, was later calculated by Kroll and Lamb, as well as French and Weisskopf This significant contribution to quantum electrodynamics has been qualitatively discussed, emphasizing its importance in the effective Dirac equation framework.
In the analysis of kernels involving spanned Coulomb photons, it is essential to consider diagrams where radiative photons span any number of exchanged Coulomb photons, as illustrated in Figures 2.1 and 3.2 The primary logarithmic contribution to the Lamb shift arises from the slope of the Dirac form factor F1, yet all these kernels can yield corrections on the order of α(Zα)⁴, making it crucial not to overlook any of them Additionally, the presence of infrared divergences in the kernels on-shell poses challenges; however, true infrared divergences are absent in the bound state problem, as they are effectively cut off by either the inverse Bohr radius or the electron binding energy Despite this, the existence of spurious on-shell infrared divergences can complicate the calculations involved.
To effectively address various treatment challenges, it is crucial to separate the radiative photon integration region using the auxiliary parameter σ, ensuring that m(Zα)²σm(Zα) In the high momentum region, each additional Coulomb photon introduces an extra factor of Zα, allowing us to focus solely on the kernel with one Coulomb photon The auxiliary parameter σ also serves as an infrared cutoff for the vertex graph, effectively mitigating potential infrared divergence issues By selecting σ as m(Zα)², the binding energy can be disregarded in the high momentum region Ultimately, the primary contribution to the Lamb shift arises from the Dirac form factor.
The term F 1(k 2 )−1 is proportional to the squared transferred momentum at low momentum transfer, which cancels the Coulomb photon propagator linked to the Dirac form factor This results in the factorization of momentum space integrations over wave function momenta, yielding the wave function squared at the origin in coordinate space Considering the small virtuality of external electron lines introduces an additional momentum squared factor in the integrand, contributing an extra factor of (Zα) 2 to the energy shift upon integration with the wave function However, for contributions of order α(Zα) 4 , the virtuality of the electron line in the high momentum region can be disregarded Notably, the high momentum region does not contribute to non-S states, as the wave function at the origin is zero for these states, leading to the absence of logarithmic contributions.
In the low momentum integration region, approximations become invalid as the influence of multiple radiative photon exchanges must be considered simultaneously, particularly for soft photons characterized by momenta of order m(Zα)² This necessitates calculating the matrix element of the exact self-energy operator within the external Coulomb field, utilizing Dirac-Coulomb wave functions While this may appear complex, it can be effectively addressed through traditional perturbation theory by incorporating a complete set of intermediate states and applying the dipole approximation, which maintains accuracy up to α(Zα)⁴m The upper boundary for the auxiliary parameter σ is strategically chosen to ensure the dipole approximation's validity Importantly, different approximations can be employed for high- and low-momentum contributions, with the high-momentum region allowing for small binding corrections, while the low-momentum region primarily relies on the dipole contribution via a nonrelativistic multipole expansion Notably, both expansions can be matched at k ∼ σ without sacrificing accuracy, leading to the classical result for the energy level shift induced by the slope of the Dirac form factor.
The reduced mass, denoted as \( m_r = \frac{mM}{m+M} \), plays a crucial role in atomic energy levels, which are influenced solely by this mass in the nonrelativistic approximation The Bethe logarithm, represented as \( \ln k_0(n, l) \), is significant in this context The factor \( \frac{m}{m_r} \) appears in the logarithm \( \ln\left(\frac{m}{\lambda}\right) \), where the infrared divergence is effectively regulated by the binding energy \( m_r (Z\alpha)^2 \).
The mass dependence of the correction of order α(Zα)⁴ is accurately represented by the expression in (3.7), as demonstrated in previous studies [11, 12] Similar to the leading relativistic correction in (3.4), the result in (3.7) remains exact for the small mass ratio m/M, since all order (Zα)⁴ corrections arise from kernels involving one-photon exchange within the effective Dirac equation framework Earlier research expanded the reduced mass factors in (3.7) to the first order in the small mass ratio; however, it is crucial to maintain the exact mass dependence in (3.7) due to the potential for current experiments to detect quadratic mass corrections, which could affect the leading nonrecoil Lamb shift contribution by approximately 2 kHz for the 1S level in hydrogen.
The Bethe logarithm is defined as a normalized infinite sum of matrix elements of the coordinate operator applied to the Schrödinger-Coulomb wave functions This pure number can be calculated with high precision, and numerous accurate results for the Bethe logarithm are available in the literature For easy reference, some values for the Bethe logarithms have been compiled in Table 3.1.
The Pauli form factor F2 contributes to the Lamb shift, but its impact is negligible if the lower components of the unperturbed wave functions are ignored, as the matrix element between the upper components is zero in the standard Dirac matrix representation However, when considering the lower components in a nonrelativistic approximation, we can derive a clear expression for the perturbation, δV = −1.
In the context of the Coulomb potential, represented as V = −Zα/r, an additional factor m/m_r appears before the second term due to radiatively corrected one-photon exchange In momentum space, the Laplacian term of the Coulomb potential relies solely on the exchanged momentum, while the second term explicitly incorporates the electron momentum Notably, the Pauli form factor is dependent on the electron momentum rather than the relative momentum of the electron-proton system This transition to relative momentum, which is relevant for the unperturbed wave functions, results in the emergence of the extra factor m/m_r.
The interaction potential above generated by the Pauli form factor may be written in terms of the spin-orbit interaction δV Zαπ m 2 δ 3 ( r ) + Zαπ r 3 mm r
The respective contributions to the Lamb shift are given by
Table 3.1.Bethe Logarithms for Lower Levels [14] lnk 0(n, l) ∆E=− 4 3 lnk 0(n, l) α(Zα) πn 3 4 ( m m r ) 3 mkHz
3.2 Radiative Corrections of Orderα n (Zα) 4 m 27 where we have used the Schwinger value [15] of the anomalous magnetic mo- mentF 2(0) =α/(2π) Correct reduced mass factors have been retained in this expression instead of expanding inm/M.
The primary contribution of the polarization operator to the Lamb shift, as illustrated in Fig 2.2, has been previously calculated in equation (2.6) By reintroducing the reduced mass factors that were excluded in the earlier qualitative analysis, we can straightforwardly derive the results.
The original result for the 2S−2P splitting, established long before modern Quantum Electrodynamics (QED), served as the sole known source for this phenomenon Interestingly, for many years, it was widely believed that this effect was too small and had the incorrect sign to account for the 2S−2P splitting.
Radiative Corrections of Order α 2 (Zα) 4 m
The theoretical calculation of corrections of order α²(Zα)⁴ does not introduce fundamentally new concepts compared to the corrections of order α(Zα)⁴ The scale for these corrections is determined by the factor 4α²(Zα)⁴/(π²n³)m, as discussed in Section 2.3 These corrections rely solely on the values of the form factors and their derivatives at zero transferred momentum, with the primary challenge being to accurately compute the corresponding radiative corrections.
The calculation of the order α²(Zα)⁴ contribution from radiative photon insertions in the electron line is more straightforward than that of the leading order contribution This is because the second and higher order contributions to the slope of the Dirac form factor are infrared finite Consequently, the total order (Zα)⁴ contribution to the Lamb shift can be directly derived from the slope of the Dirac form factor, eliminating the need to sum an infinite number of diagrams The respective contribution can be easily obtained.
28 3 External Field Approximation where we have used the second order slope of the Dirac form factor generated by the diagrams in Fig 3.3 dF 1 (2) (−k 2 ) dk 2 |k 2 =0
The early pioneering works on the two-loop slope laid the groundwork for further research, with the first accurate numerical result achieved in [19] This breakthrough spurred a significant increase in theoretical activity, as seen in works [20, 21, 22, 23] Subsequently, the first fully analytical calculation was presented in [24], while an equivalent analytical result for the slope of the Dirac form factor was derived from the total e + e − cross section and the unitarity condition in [25].
The calculation of the Pauli form factor contribution closely resembles the previous work conducted for α(Zα)⁴, with the key distinction being the use of the second-order contribution to the Pauli form factor This calculation, illustrated in Fig 3.3, was established in earlier studies [26, 27, 28].
[26] turned out to be in error)
(3.15) Then one readily obtains for the Lamb shift contribution
Fig 3.3.Two-loop electron formfactor
Fig 3.4.Insertions of two-loop polarization operator
Here we use well known low momentum asymptotics of the second order po- larization operator [29, 30, 31] in Fig 3.4 Π(−k 2 ) k 4 |k 2 =0
Corrections of Order α 3 (Zα) 4 m
The calculation of the α³(Zα)⁴ corrections mirrors that of the α²(Zα)⁴ contributions, with both relying solely on the three-loop form factors or their derivatives at zero transferred momentum The three-loop contribution to the slope of the Dirac form factor has been analytically determined, as indicated in the referenced study.
The respective contribution to the Lamb shift is equal to
The calculation of the Pauli form factor contribution to the Lamb shift utilizes the third-order contribution to the Pauli form factor, which has been computed both numerically and analytically in previous studies.
Then one obtains for the Lamb shift
Fig 3.5.Examples of the three-loop contributions for the electron form factor
In this case the analytic result for the low frequency asymptotics of the third order polarization operator (see Fig 3.6) [35] is used
Fig 3.6.Examples of the three-loop contributions to the polarization operator Π(−k 2 ) k 4 |k 2 =0
Total Correction of Order α n (Zα) 4 m
The total contribution of orderα n (Zα) 4 mis given by the sum of corrections in (3.7), (3.11), (3.12), (3.13), (3.16), (3.18), (3.21), (3.23), (3.25) It is equal to
32 3 External Field Approximation for theS-states, and
Numerically corrections of order α n (Zα) 4 m for the lowest energy levels give
Contributions of order α^4 (Zα)^4 m are significantly suppressed by an additional factor of α/π compared to the corrections of order α^3 (Zα)^4 m The anticipated magnitude of these contributions is around hundredths of kHz, even for the 1S state in hydrogen, making them too small to have any phenomenological significance.
Heavy Particle Polarization Contributions
In addition to virtual photons and electrons, it is essential to consider the effects of virtual muons and the lightest strongly interacting particles for accurate calculations The known corrections to the electron anomalous magnetic moment are minimal and do not significantly impact Lamb shift calculations Notably, the contributions from heavy particles to the polarization operator are comparable in magnitude to polarization corrections of order α³(Zα)⁴ Furthermore, low-frequency asymptotic corrections to the polarization operator arise from specific diagrams, highlighting the importance of the muon loop contribution.
Fig 3.7 Muon-loop and hadron contributions to the polarization operator Π(−k 2 ) k 4 |k 2 =0
15πm 2 à (3.29) immediately leads (compare (3.12)) to an additional contribution to the Lamb shift [37, 38]
The hadronic polarization's impact on the Lamb shift has been explored in several studies Estimating the light hadron contribution to the polarization operator can be effectively achieved using vector dominance, represented as Π(−k²) k⁴ |k²=0.
The total hadronic vacuum polarization contribution to the Lamb shift can be expressed using the formula 4πα f v² i m² v i, where m v i represents the masses of the three lowest vector mesons The vector meson-photon vertex is characterized by the form em² v i /f v i, and this analysis incorporates free quark loops for the contribution of heavy quark flavors.
Numerically this correction is−3.18 kHz for the 1S-state and−0.40 kHz for the 2S-state in hydrogen.
The leading coefficient in the low energy expansion of hadronic vacuum polarization is essential for calculating the hadronic contribution to the Lamb shift A model-independent value for this coefficient can be derived from the analysis of experimental data on low energy e+ e− annihilation The corresponding contribution to the 1S Lamb shift is −3.40(7) kHz, which is both compatible and more precise than previous results.
All corrections of orderα n (Zα) 4 mare collected in Table 3.2.
The inclusion of hadronic vacuum polarization in the phenomenological analysis of Lamb shift measurements is not immediately apparent, as it is experimentally indistinguishable from an additional factor affecting the proton charge radius This issue will be further explored in Section 6.1.3.
Two-loop Pauli FFl= 0 ( 16 3 ζ(3)− π 8 2 ln 2 + π 48 2 + 197 576 ) α π
Dirac FF − 217 216 ln 4 2− 1080 103 π 2 ln 2 2 + 41671 2160 π 2 ln 2
Kinoshita (1990) [33] + 17 101 3 240 π 2 + 28 259 20 736 } j(j+1) l(l+1)(2l+1) − l(l+1) − 3/4 m m r ( α π ) 2 Laporta, Remiddi (1996) [34] ≈0.295 310 3 j(j+1) l(l+1)(2l+1) − l(l+1) − 3/4 m m r ( α π ) 2 Three-loop Vacuum Polarization
Radiative Corrections of Order α n (Zα) 5 m
Skeleton Integral Approach to Calculations
We have seen above that calculation of the corrections of order α n (Zα) 4 m
The study of higher order corrections to the properties of free electrons and photon propagators focuses on calculating the slope of the electron Dirac form factor, the anomalous magnetic moment, and the leading term in the low-frequency expansion of the polarization operator These contributions to the Lamb shift are notably independent of bound state characteristics A complex interaction between radiative corrections and binding effects emerges when examining contributions of order α(Zα)⁵m and in higher order calculations within the combined expansion of α and Zα.
The calculation of the order α n (Zα) 5 m contribution to the energy shift is simpler than that of the leading order contribution to the Lamb shift, as the scattering approximation suffices This correction arises from kernels involving at least two-photon exchanges, but unlike the leading order, higher-order irreducible kernels do not appear This is because, in the high exchanged momentum region, the expansion in Zα is valid, and adding extra exchanged photons yields additional powers of Zα, making only two exchanged photon diagrams relevant Furthermore, the infrared behavior of radiatively corrected Feynman diagrams is milder than that of the skeleton diagram, simplifying the treatment of the low-momentum region In momentum space, the matrix element of the diagram with two exchanged Coulomb photons between the Schrödinger-Coulomb wave functions reveals that the contribution to the Lamb shift is represented by an infrared divergent integral.
Fig 3.8 Skeleton diagram with two exchanged Coulomb photons
Radiative insertions in the electron line, as depicted in Fig 3.9, involve a dimensionless momentum \( k \) of the exchanged photon, measured in units of the electron mass This divergence can be physically interpreted by considering the small virtualities of the external electron lines and wave functions; the two-Coulomb exchange effectively adds an extra rung to the Coulomb wave function, which should reproduce it The naive infrared divergence is regularized at the characteristic atomic scale \( mZ\alpha \), indicating that the two-photon exchange kernel is already incorporated in the effective Dirac equation, negating the need for perturbative consideration Furthermore, accounting for radiative photon insertions results in an additional factor \( L(k) \) in the divergent integral, which exhibits logarithmic asymptotic behavior at high momenta, while in the low momentum region, it behaves as \( L(k) \sim k^2 \).
The inclusion of radiative corrections improves the low-frequency behavior of the integrand, yet it remains divergent due to contributions from the two-photon-exchange box diagram, specifically the leading term related to the Lamb shift induced by the electron form factor To address this, the leading low momentum term can be subtracted from L(k)/k^4, resulting in a convergent integral that accounts for the correction of order α(Zα)^5 This method also conveniently eliminates concerns about ultraviolet divergence, ensuring that the outcome is both ultraviolet and infrared finite.
Radiative insertions lead to the suppression of low integration momenta in exchange loops, resulting in effective integration momenta of order m Consequently, the small virtuality of external fermion lines can be ignored, allowing calculations to be performed using on-mass-shell external momenta The contributions to the Lamb shift are determined by the product of the square of the Schrödinger-Coulomb wave function at the origin, |ψ(0)|², and the corresponding diagram Under these conditions, the diagrams presented in Fig 3.9 form a gauge-invariant set that can be easily calculated.
Higher-order contributions involving diagrams with more than two exchanged Coulomb photons are significant in the context of Zα This is particularly evident in the integration region of high exchanged momenta Additionally, it can be shown that in the Yennie gauge, the contributions from low exchanged momentum also play a role.
The external field approximation for the matrix element with on-shell external electron lines remains infrared finite, indicating that it cannot yield corrections of order α(Zα)⁵ This conclusion holds true across all gauges due to the gauge invariance of the sum of diagrams involving on-shell external electron lines Additionally, the small virtuality of these external electron lines further suppresses the matrix element, making it sufficient to focus solely on two-photon exchanges when calculating all corrections of order α(Zα)⁵.
The correction magnitude of order α(Zα) 5 can be preliminarily estimated by considering the skeleton factor 4m(Zα) 4 /n 3 mentioned in Section 2.3, and multiplying it by an additional factor of α(Zα) One might naively anticipate a slightly reduced factor of α(Zα)/π.
A convergent diagram involving two external photons introduces an additional factor of π in the numerator, which balances the π factor in the denominator due to radiative corrections Consequently, the calculation of the correction of order α(Zα)⁵ results in a numerical factor of order unity, multiplied by 4mα(Zα)⁵/n³.
Radiative Corrections of Order α(Zα) 5 m
3.3.2.1 Correction Induced by the Radiative Insertions in the Electron Line
The correction arises from the total contributions of all potential radiative insertions in the electron line, as illustrated in Fig 3.9 To obtain the electron factor reflecting the cumulative radiative corrections, it is essential to subtract the leading infrared asymptote This adjusted expression is then incorporated into the integrand of equation (3.33), followed by integration over the exchanged momentum, ultimately yielding the desired result.
3 m δ l0 , (3.34) which was first obtained in [40, 41, 42] in other approaches Note that numer- ically 1 + 11/128−1/2 ln 2≈0.739 in excellent agreement with the qualitative considerations above.
3.3.2.2 Correction Induced by the Polarization Insertions in the External Photons
The correction of order α(Zα) 5, resulting from the insertion of the polarization operator in the external photon lines as depicted in Fig 3.10, has been previously derived in references [40, 41, 42] This correction can also be computed using the skeleton integral approach To illustrate the general considerations discussed earlier, we will focus on the straightforward one-loop polarization operator and conduct a detailed calculation This involves inserting the polarization operator into the skeleton integrand as specified in equation (3.33).
Fig 3.10.Polarization insertions in the Coulomb lines
Of course, the skeleton integral still diverges in the infrared after this substi- tution since
The linear infrared divergence of the term dk/k² is effectively mitigated at the atomic scale of mZα This reduction decreases the power of the factor Zα, resulting in a divergent contribution that is of the order of α(Zα)⁴ This adjustment corresponds to the polarization component of the leading order contribution to the Lamb shift.
We perform the subtraction of the leading low-frequency asymptote from the polarization operator insertion, which is essential for adjusting the integrand related to the energy shift contribution.
To analyze the Lamb shift, we substitute the expression \(4 + (1-v^2)k^2\) into the formula presented in equation (3.33) Additionally, we incorporate a factor of 2 to account for potential insertions of the polarization operator in both photon lines.
In our analysis, we have reintroduced the characteristic factor 1/(1−m²/M²) that was previously omitted, as it naturally appears in the skeleton integral However, the impact of this omission is minimal, resulting in an error of only approximately 0.02 kHz for the electron-line contribution to the 1S level shift Therefore, this correction can be disregarded given the current level of experimental accuracy.
The total correction of order α(Zα) 5 m is given by the sum of contributions in (3.34), (3.39)
Corrections of Order α 2 (Zα) 5 m
Corrections of order α²(Zα)⁵ share the same physical origin as those of order α(Zα)⁵, and their calculation can be effectively approached using the scattering approximation In this context, we explore higher-order corrections in α, which involve a broader range of relevant diagrams Figure 3.11 illustrates all six gauge-invariant sets of diagrams that yield corrections of order α²(Zα)⁵ Notably, the "2 loops" blob in Figure 3.11 (f) represents the gauge-invariant sum of diagrams featuring all possible insertions of two radiative photons along the electron line These diagrams can be derived from the skeleton diagram presented in Figure 3.8 through various two-loop radiative insertions Similar to the corrections of order α(Zα)⁵, the energy shift corrections are determined by the matrix elements of the diagrams in Figure 3.11, computed between free electron spinors with all external electron lines on-shell, projected onto their respective spin states, and multiplied by the square of the Schrödinger-Coulomb wave function at the origin.
Some diagrams include contributions from the previous order in Zα, resulting from terms proportional to the squared exchanged momentum in the low-frequency asymptotic expansion of radiative corrections These contributions are linked to the integration over external photon momenta at the characteristic atomic scale mZα, making the scattering approximation insufficient for their calculation In the skeleton integral approach, these previous order contributions manifest as powerlike infrared divergences during the final integration over exchanged momentum To eliminate these spurious contributions, we subtract the leading low-frequency terms from the asymptotic expansions of the integrands when necessary.
3.3.3.1 One-Loop Polarization Insertions in the Coulomb Lines
The simplest correction is induced by the diagrams in Fig 3.11 (a) with two insertions of the one-loop vacuum polarization in the external photon lines.
Fig 3.11.Six gauge invariant sets of diagrams for corrections of orderα 2 (Zα) 5 m
The Lamb shift is influenced by the insertion of the one-loop polarization operator squared I 1 2 (k) within the skeleton integral outlined in equation (3.33) By considering the multiplicity factor of 3, one can readily derive the results.
3.3.3.2 Insertions of the Irreducible Two-Loop Polarization in the Coulomb Lines
The naive insertion of the irreducible two-loop vacuum polarization operator I²(k) into the skeleton integral results in an infrared divergent integral, as demonstrated in the diagrams of Fig 3.11 (b) This divergence indicates the presence of corrections from the previous order in Zα, associated with the two-loop irreducible polarization The contribution of order α²(Zα)⁴m was previously discussed in Subsect 3.2.2.3, and it was established that the corresponding contribution to the Lamb shift is determined by the product of the squared Schrödinger-Coulomb wave function at the origin and the leading term.
The low-frequency term of the function I 2(0) in the external field approximation indicates that the relevant loop momenta are on the atomic scale, represented as mZα By subtracting I 2(0) from I 2(k), the contribution from the low momentum region is effectively removed from the loop integral This process yields the radiative correction of order α² (Zα)⁵, which is generated by the irreducible two-loop polarization operator.
3.3.3.3 Insertion of One-Loop Electron Factor in the Electron Line and of the One-Loop Polarization in the Coulomb Lines
The correction of order α² (Zα) 5 is derived from a gauge-invariant set of diagrams, as illustrated in Fig 3.11 (c) To obtain the corresponding analytic expression, we perform a simultaneous insertion of the one-loop polarization function I₁(k) along with all possible radiative photon insertions in the electron line within the skeleton integral To simplify the process, we first derive an explicit analytic expression for the total of these radiative insertions in the electron line, referred to as the one-loop electron factor L(k).
(explicit expression for the electron factor in different forms may be found in
The electron factor is first calculated and then inserted into the skeleton integral, resulting in an integral for the radiative correction that is both ultraviolet and infrared finite This infrared finiteness aligns with the physical understanding that there is no correction of order α2 (Zα)4 generated at the atomic scale The integral for the radiative correction has been calculated both numerically and analytically, yielding an elegant result that confirms the theoretical expectations.
3.3.3.4 One-Loop Polarization Insertions in the Radiative Electron Factor
The correction arises from the gauge invariant diagrams shown in Fig 3.11(d), which include polarization operator insertions in the radiative photon The formula for the radiatively corrected electron factor is provided in reference [50].
1−v 2 L(k, λ), (3.44) where L(k, λ) is just the one-loop electron factor used in (3.43) but with a
The direct substitution of the radiatively corrected electron factor L(k) into the skeleton integral results in an infrared divergence, which indicates the presence of a previous order correction in Zα due to two-loop insertions in the electron line This divergence is influenced by the nonzero value of the electron factor L(k) at zero, determining the magnitude of the prior order correction.
2F 2(0), (3.45) which is simply a linear combination of the slope of the two-loop Dirac form factor and the two-loop contribution to the electron anomalous magnetic moment.
Subtraction of the radiatively corrected electron factor removes this pre- vious order contribution which was already considered above, and leads to a finite integral for the correction of orderα 2 (Zα) 5 [50, 47]
3.3.3.5 Light by Light Scattering Insertions in the External Photons
The diagrams in Fig 3.11 (e) with the light by light scattering insertions in the external photons do not generate corrections of the previous order inZα.
Ultraviolet and infrared calculations are finite and, while technically complex, are conceptually straightforward Numerical results have been derived for the contributions to the Lamb shift.
3.3.3.6 Diagrams with Insertions of Two Radiative Photons in the Electron Line
Diagrams featuring radiative insertions in the electron line consistently dominate the contributions compared to those with radiative insertions in the external photon lines This characteristic highlights the significance of electron line interactions in the analysis.
The external field approximation in quantum electrodynamics (QED) relies on gauge invariance, ensuring that diagrams with external photon lines remain gauge invariant through the use of transverse projectors These projectors, which are rational functions of external momenta, effectively suppress low momentum integration regions in energy shift calculations In contrast, diagrams featuring insertions in the electron line lack these projectors, resulting in less suppression of low momentum regions and consequently leading to larger contributions to energy shifts.
Radiative corrections are prominently illustrated by six gauge invariant sets of diagrams, particularly highlighted in Fig 3.11 The most significant contribution to the Lamb shift, of order α²(Zα)⁵, arises from the last set of diagrams in Fig 3.11 (f), which includes nineteen unique topological diagrams as shown in Fig 3.12 These diagrams can be derived from the two-loop electron self-energy graphs by inserting two external photons in various configurations Specifically, diagrams in Fig 3.12 (a–c) stem from the two-loop reducible electron self-energy, while those in Fig 3.12 (d–k) result from photon insertions in the rainbow self-energy diagram, and diagrams in Fig 3.12 (l–s) relate to the overlapping two-loop self-energy graph Initial calculations of the energy shift were conducted in previous studies, which focused on diagrams from Fig 3.12 (a–h) and Fig 3.12 (l) The comprehensive contribution of all nineteen diagrams to the Lamb shift was first calculated in subsequent research, with further analysis completed through the skeleton integral approach.
3 m δ l0 , (3.48) which confirmed the one in [55] but is about two orders of magnitude more precise than the result in [55, 57].
The significance of this result is often overstated in the literature, but a quick review of Table 3.3 reveals otherwise For those who have closely followed the discussion on various contributions, it is evident that the appropriate scale for the correction in question is determined by the factor 4α²(Zα)⁵/(πn³)m The coefficient derived in (3.48) is approximately -1.9, which is typical for a numerical factor related to a radiative correction In comparison, the coefficient of 0.739 is noted before the factor 4α(Zα)⁵/n³m for the electron-line contribution from the previous order in α.
The misconception regarding the correction of order α²(Zα) 5m stems from the belief that the energy expansion series, when expressed in terms of the fixed power of α, should feature coefficients of order one However, as highlighted in previous discussions, this assumption is not accurate.
Corrections of Order α 3 (Zα) 5 m
Corrections of order α³ (Zα)⁵ are analogous to those of order α² (Zα)⁵ and can be computed using a similar method Their natural scale is defined by the factor 4α³ (Zα)⁵ / (π² n³) m Considering an additional numerical factor of approximately 1−2, we estimate that these corrections are about 1 kHz for the 1S-state and around 0.1 kHz for the 2S-state.
Corrections of order α^3 (Zα)^5 arise from three-loop radiative insertions in the skeleton diagram illustrated in Fig 3.8 These corrections are associated with diagrams that include at least one one-loop or two-loop polarization insertion, as detailed in reference [58].
Based on our experience with the corrections of order α² (Zα)⁵, we anticipate that the primary contributions to the corrections of order α³ (Zα)⁵ will arise from a gauge-invariant set of diagrams featuring three insertions.
The analysis focuses on the three-loop one-particle reducible diagrams featuring radiative photons along the electron line, as depicted in Fig 3.8 and Fig 3.13 In the Yennie gauge, the contributions from these specific diagrams have been evaluated, highlighting their significance in the context of external field approximation.
Work on calculation of the remaining contributions of orderα 3 (Zα) 5 is in progress now.