BPS Explained I: Temperature Relaxation in a Plasma or How to Find the Coulomb Logarithm Exactly

BPS Explained I: Temperature Relaxation in a Plasma or How to Find the Coulomb Logarithm Exactly

Abstract

This is the first of two lectures on the technique of dimensional continuation employed by Brown, Preston, and Singleton (BPS) to calculate such quantities as the charged particle stopping power and the temperature equilibration rate in a plasma. In this exposition we will examine some of the more basic points of dimensional continuation, with an emphasis on the Coulomb logarithm for electron-ion temperature equilibration. Dimensional continuation, or dimensional regularization as it is more properly known in quantum field theory, was originally developed as part of the renormalization procedure for the theories of the electroweak and other fundamental interactions in particle physics. Dimensional continuation is so general, in fact, that any theory can be unambiguously lifted to dimensions beyond three, and therefore the technique is powerful enough to apply in many other settings. The technique, however, is not well known outside the field theory and particle physics communities. This exposition will therefore be self-contained, intended for those who are not specialists in quantum field theory, and I will either derive or motivate any requisite field theory results or concepts. Of particular relevance is the analogy between the Coulomb logarithm as calculated by Lyman Spitzer on the one hand, and the Lamb shift as calculated by Hans Bethe on the other. While dimensional continuation is a well developed and a thoroughly tested method for regularizing any quantum field theory, BPS employs the method in a novel way that provides the leading and subleading behavior for processes that involve competing disparate energy or length scales. In particular, BPS calculated the temperature equilibration rate to leading and next-to-leading order in the plasma number density for any two species in a plasma that are in thermal equilibrium with themselves, but not necessarily with each other. No restriction is made on the charge, mass, or the temperature of the plasma species. It is, however, assumed that the plasma is not strongly coupled in the sense that the dimensionless plasma coupling parameter is small, where is the Debye wave number of the plasma. To leading and next-to-leading order in this coupling, the temperature equilibration rate is of the generic form . The precise numerical coefficient in front of the logarithm has been known for some time, while BPS have recently computed the constant under the logarithm. It should be emphasized that the BPS result is not a model, but rather it is an exact calculation of the leading terms in a well-defined perturbation theory. This exact result differs from approximations and models given in the literature.

1

I Introduction and Context

This is the first of two lectures on a new technique for calculating the temperature equilibration rate between electrons and ions in a weakly to moderately coupled fully-ionized plasma, exact to leading and next-to-leading order in the plasma number density. This calculation was first performed in Section 12 of Ref. (1), a work whose primary focus was the charged particle stopping power in a plasma.2 This paper assumed familiarity with a number of field theory concepts, and Section 12 relied heavily on the charged particle stopping power results derived in previous sections of that work. In contrast, these lectures will be self-contained. I will either derive or motivate the requisite field theory background for a complete reading of Ref. (1), with an emphasis on some of the subtleties of the calculational techniques and the concepts behind them (this lecture). More to the point, since the rate calculation stands on its own, it should be presented on its own (the following lecture). Indeed, since the calculation of the rate is somewhat less involved than that of the stopping power, it more clearly illustrates the tools and concepts imported from field theory.

In addition to clarifying the method of dimensional continuation, this lecture will place Ref. (1) in the context of more familiar and traditional approaches to the rate problem. In particular, I will show that dimensional continuation can be viewed as a systematic implementation of the approach based on convergent kinetic equations. Finally, in in the next lecture, I will go on to derive the main result from Section 12 of Ref. (1), the rate coefficient (3) of this lecture. By working in the Born approximation, and adopting the methods of Ref. (3), I will derive this result in a much simpler manner than originally presented in Ref. (1). While Lectures I and II are self-contained, they are complementary and should be read as a unit.3

The strategy employed by Ref. (1), hereafter referred to as BPS, consists of two steps. First, we will find a dimensionless parameter in which to perform a controlled perturbative expansion of the rate, expanding to leading and next-to-leading order in this parameter. Second, we will deploy a technique from quantum field theory that will allow us to calculate the coefficients of these leading and subleading terms exactly. The exact leading order term is not very difficult to find, and has been known since the classic work of Spitzer. The next-to-leading order term, on the other hand, was not known exactly until the recent BPS calculation. The third-order term provides an estimate of the error of the calculation. When the plasma is weakly to moderately coupled, the error will be small and the rate will be approximated quite accurately by the first two terms of this expansion.

To calculate the expansion coefficients, we will exploit a field theory technique known as dimensional regularization (or dimensional continuation, as I will call it here). This application of dimensional continuation is quite different from its intended purpose in the renormalization procedure of quantum field theory. Dimensional continuation was originally developed as an elegant regularization scheme in which the fundamental symmetries of a field theory could be maintained while still rendering finite the otherwise infinite integrals that arise when calculating Feynman diagrams. I will show how this technique can be used in a novel fashion to extract next-to-leading order physics that has, until now, remained inaccessible. In other words, I will show how dimensional continuation provides an exact result for the corresponding Coulomb logarithm of the process in question. I will also take the opportunity to correct a small algebra mistake for the electron-ion equilibration rate presented in Section 12 of Ref. (1).

i.1 The Problem

The general formalism starts with a plasma composed of multiple species labeled by an index , the various species being delineated by of a common electric charge and a common mass .4 Each species is assumed to be in thermal equilibrium with itself at temperature with a spatially uniform number density . I will drop the subscript on the charge of the electron and write (with ), although the electron mass will be denoted by , and the number density and temperature of the electron plasma component by and respectively. I will use a lower case subscript to denote a single ion species of charge , mass , number density , and temperature . I will employ a capital-I subscript to denote properties that correspond to the collective set of ions, such as the total ion number density or a common ion temperature . When equilibrium distributions are required in calculations, they are assumed to be Maxwell-Boltzmann, although a generalization to Fermi-Dirac statistics can be accomplished with more effort (3). For problems involving hot thermonuclear burn, however, the fugacity is small and Maxwell-Boltzmann statistics is an accurate approximation.

Let denote rate at which the energy density of plasma species changes because of its Coulomb interactions with another species (the rate from the -species to the -species). This rate is proportional to the temperature difference, and can be expressed by

(1)

The sign convention in (1) implies that when the rate coefficients are positive, then energy will flow from the hotter species to the cooler species, as it must. Section 12 of BPS used dimensional continuation to calculate the general rate coefficients in a weakly coupled, but otherwise arbitrary, plasma. For simplicity, we will not perform the general calculation until Lecture III. In this and the following lecture, we will concentrate on the energy exchange between electrons and ions only. Since the electron mass is so much smaller than a typical ion mass , the electrons will come into equilibrium first with temperature on some time scale . The energy transfer rate among ions is a factor slower than the corresponding rate for electrons, and therefore the ions will equilibrate to a common temperature in a time  . Finally, as the electrons and ions exchange energy through Coulomb interactions, these systems too will equilibrate on a time scale . Consequently, one finds a hierarchy of time scales , and it indeed makes sense to consider the electron and ion systems as having distinct temperatures and , with subsequent equilibration between them. Taking and in (1), and since the ions have a common temperature , the rate equation of interest is obtained by summing over the ion components of (1) to give

(2)

where and .

The coefficient is the quantity we wish to calculate in this and the next lecture. This coefficient contains the energy-exchange physics between electrons and ions resulting from mutual Coulomb interactions, including possible collective effects and large-angle collisions. General expressions for the individual were calculated in Section 12 of BPS. They are somewhat complicated and involve various one-dimensional integrals that can only be performed numerically. However, the collective rate coefficient simplifies considerably when the mild restriction   is imposed (a sum-rule is employed in the approximation, and the simplification occurs only for and not for the individual ). If the high temperature limit is further imposed, then the rate can be written in a quite simple analytic form.

(3)

where is the Euler constant, and are the electron Debye wave number and plasma frequency, and is sum of the squares of the ion plasma frequencies. Since the small binding energy of the hydrogen atom sets the temperature scale, and since the condition is not very restrictive, the rate coefficient (3) is applicable in almost all circumstances of interest. We shall devote the next lecture to deriving this expression. For now, note that equation (3) corresponds to Eqs. (3.61) and (12.12) of Ref. (1), where I have taken this opportunity to correct a small transcription error: when passing from Eq. (12.43) to Eq. (12.44) in Ref. (1), a factor of 1/2 was dropped. Restoring this factor of 1/2 changes the additive constant outside the logarithm from the that appears in Eq. (12.12) of Ref. (1) to the constant in (3) above.

For reasons to be discussed shortly, rationalized units are preferred for dimensional continuation, and I will employ this choice in all that follows. Nonetheless, expression (3) is written in a manner that does not depend upon this choice: the Debye wave number , and the plasma frequencies and can be calculated in your favorite units. For example, in Gaussian units where the electric potential takes the form , the Debye wave number and the plasma frequency of species are given by and . In the rationalized units employed here, the electric potential is given by , and we have

(4)
(5)

The square of the total Debye wave number is , and the total Debye wave length is .

i.2 The Problem with the Problem

Let us now consider an arbitrary plasma component of mass , which I will otherwise leave unspecified, and let denote the Boltzmann distribution for this species. Then the average (kinetic) energy density of this component is

(6)

If we work to leading and next-to-leading order in the number density, calculating the energy exchange between plasma components will then involve keeping a tally only of the kinetic energy, as in (6). This is because the potential energy is higher order in the number density [or more precisely, the potential energy is higher order in the plasma coupling , to be defined later in (36)]. As the system interacts with other plasma components through mutual Coulomb interactions, it will loose or gain energy depending on the temperature gradients with other species, and the energy exchange rate is given by

(7)

In contrast to (1) and (2), for ease of notation I have temporarily dropped the plasma component subscripts on the rate, and I will keep with this convention until the final calculation presented in Section VI.2. We see that the entire problem is bound up in calculating the rate of change from an appropriate kinetic equation that captures the relevant physics. As it turns out, however, there is a serious problem in performing all such calculation with the Coulomb potential in three dimensions: the integrals in the kinetic equations diverge logarithmically, and they do so at both large and small distances.5

For processes in which large-angle scattering is important, such as the charged particle stopping power, it is natural to use the Boltzmann equation, which I will write in the abbreviated form

(8)

where is the gradient in position space, and is the scattering kernel, whose precise form will not concern us until the next lecture. The gradient vanishes because of spatial uniformity, so we will set . The Boltzmann equation was designed to account for the statistical effects of short-distance collisions, and although its original context was classical, quantum two-body scattering effects can easily be incorporated. In fact, since the scattering phase shifts are known analytically for the Coulomb potential, Ref. (1) used this to calculate the two-body quantum corrections to all orders [in the quantum parameter to be defined in (48)]. The kernel therefore contains all short-distance or ultraviolet physics, for both classical and quantum scattering. However, (in three spatial dimensions) the Coulomb potential is long-range, and the integrals in diverge in the infrared; or equivalently, if we write the scattering kernel in terms of momentum integrals, the divergence appears at small values of momentum. This was not a problem in Boltzmann’s original formulation of (8), since the Coulomb potential was unknown at that time, and he modeled particle collisions in terms of hard-sphere scattering. In summary, the Boltzmann equation gets the short-distance physics correct, including quantum two-body scattering, but it misses the infrared physics. The fact that the Boltzmann equation misses the long-distance physics manifests itself as an infrared divergence in the scattering kernel , thereby rendering calculations meaningless (unless we tame, or regularize, this divergence).

Given that the Boltzmann equation misses the long-distance or infrared (IR) physics, we might be tempted to try the Lenard-Balescu equation, which I will write in the abbreviated form

(9)

where is a scattering kernel whose exact form will be needed only in the next lecture. Again, the gradient term will be set to zero because of spatial uniformity. The Lenard-Balescu equation takes the form of a Fokker-Plank equation, with the kernel chosen to capture the correct IR physics. However, for the Coulomb potential (in three spatial dimensions), the Lenard-Balescu equation misses the short-distance or ultraviolet (UV) physics, and this is manifested by a UV divergence in . The situation for the Lenard-Balescu equation is exactly reversed compared to that of the Boltzmann equation. This is what Ref. (4) calls the “complementarity” of these two kinetic equations, and in the dimensional continuation procedure we will use this fact to our advantage.

Ii Traditional Methods

ii.1 Heuristic Models

The rate equation (7) reduces to a one-dimensional integral over the entire range of physical length scales (or momentum scales, if one so chooses), from zero all the way to infinity. Trouble arises for the Coulomb potential in three dimensions since the integral in question is logarithmically divergent at both integration limits. We must therefore regulate the integral in some manner. Dimensional continuation is one such procedure, but there are others. This divergence problem was first worked around by simply cutting off the divergent integrals by hand, with the cutoffs themselves being chosen by physical arguments (rather than a calculation).

The energy exchange rate we are considering is but an example of a larger class of problems involving characteristic, but disparate, length or energy scales in which the measured quantity of interest is (logarithmically) insensitive to the physics above and below these scales. For these problems, the simplest and most intuitive regularization scheme is to replace the offending integration limits, i.e. infinity and zero, by the finite and non-zero physical scales of the problem. These two scales then act as formal integration cutoffs, giving a finite logarithm of the ratio of the scales. Furthermore, because the system is insensitive to the physics above and below the respective cutoffs, this procedure provides a physically meaningful result. Expressed in terms of length, we will denote the long- and short-distance scales by and , and the integral over scales leads to a finite logarithm involving the ratio of the physical length scales, so that

(10)

where is an easily determined prefactor with dimensions of energy density per unit time. As we shall see, a calculation to leading order in the number density is sufficient to provide the coefficient , while a next-to-leading order calculation is required to find the exact terms under the logarithm.

The problem with this regularization prescription, which I will call the heuristic scheme, is that we can only estimate the values of the physical scales and to within factors of order one or so. For example, it is physically reasonable that the scale of the long distance cutoff in a plasma is set by a Debye length, so that with being a dimensionless constant of order unity; but what determines the exact value of  ? In fact, how does arise naturally from the kinetic equations themselves, rather than simply being chosen by hand? And should one use the total Debye wave number , or just the contribution from the electrons ? The origin of the short distance cutoff is even less clear. In the extreme classical limit, we expect this scale to be set by the classical distance of closest approach between two colliding particles, so that .6 On the other hand, we expect quantum effects to dominate when becomes smaller than some thermal Compton wavelength , with being a typical thermally averaged momentum transfer. In this case we expect . Worse yet, the intermediate region in which neither classical nor quantum processes dominate is often realized in a weakly coupled plasma, and in this case it is even more obscure how one should choose . We must interpolate between the extreme classical and the extreme quantum scales, but in the literature the exact procedure is always somewhat ad hoc. For example, a common choice is to define to be the sum of the squares of the classical and quantum scales, so that

(11)

I will have more to say about this in Section IV.1.3.

The heuristic scheme forces us to choose the specific forms of and motivated by imprecise physical arguments or heuristic exercises, which leads us into the art of model building rather than systematic calculation. Indeed, the very notation that we must choose a cutoff is misleading, since the physics itself must conspire to render all integrals finite. Consequently, the heuristic method suffers from an unknown coefficient under the logarithm, and only the approximate value of the ratio can be determined with this method (in fact, this ratio varies across an order of magnitude over models in the literature, rather than factors of two or three). As we shall see, determining the constant under the logarithm exactly is equivalent to determining the next-to-leading order term exactly; therefore, models of the form (10) are accurate only to leading order, and no better.

ii.2 Convergent Kinetic Equations: Traditional Approach

Rather than merely regulating the integrals in a rate derived from the kinetic equations, as with (10), a more sophisticated approach involves regularizing the divergences in the kinetic equations themselves. In other words, the theory itself is regularized, rather than a particular quantity being calculated within the theory. The method of dimensional continuation falls into this category, albeit with somewhat more subtle mathematical machinery than traditional approaches. These approaches, of which Refs. (5); (6); (7) are good examples, are summarized and placed into a common framework by Aono in Ref. (4). As discussed in Section I.2 of this lecture, one can view the Boltzmann and Leonard-Balescu equations as providing complementary physics since they both succeed and fail in complementary regimes. The Boltzmann equation gets the short-distance physics correct, while the Leonard-Balescu equation captures the long-distance physics; conversely, Boltzmann and Leonard-Balescu miss the long- and short-distance physics, respectively. This complementarity motivates a class of kinetic equations of the form (4)

(12)

where is a carefully chosen “regulating kernel” designed to subtract the long-distance divergence of the Boltzmann equation and the short-distance divergence of the Lenard-Balescu equation. At the same time, the kernel must preserve the correct short-distance physics of the Boltzmann equation and the correct long-distance physics of the Lenard-Balescu equation (a minimal requirement of the regulating kernel is that it take a Hippocratic Oath to do no harm, at least to subleading order in the plasma coupling). Each term on the right-hand-side of (12) is separately divergent, but collectively they lead to a finite collision kernel if properly interpreted.7 As an example, Gould and DeWitt (7) regulated the long-distance divergence of the Boltzmann equation by simply replacing the Coulomb potential with a Debye screened Yukawa-like potential,

(13)

where, as not to confuse symbols, I write as the base of the natural logarithm and as the electric charge. While Ref. (7) performed this operation by hand using physical arguments, one could easily introduce a kernel to do the same.

While the approach to convergent kinetic equations described by Aono might appear to be more rigorous than the aforementioned model building approach of (10), it is no more systematic: methods based on (12) or its equivalent do not contain the ability to estimate their own error, i.e. they cannot determine their domain of applicability. There is nothing in the formalism of (12) that keeps track of the plasma coupling constant, or the order to which we are working in this constant. Indeed, one does not generally think of (12) in terms of a perturbation theory. In contrast, Ref. (1) calculates the rate using a systematic expansion in the plasma number density, or more precisely, in a dimensionless plasma coupling parameter [to be defined by (36) and discussed at length in Section IV.1]. Although written in a disguised form, the BPS rate coefficient (3) is an expansion to leading and next-to-leading order in the coupling parameter: the leading order term goes like , the next-to-leading order is proportional to , and the term provides an estimate of the error. Translating the work of Gould and DeWitt (7) into the language of a perturbative expansion in a plasma coupling constant, it turns out that their result is valid to order and is in agreement with BPS to this order.8 However, in their final result, Gould and DeWitt retain spurious higher order terms in . I call these terms “spurious” because Ref. (7) did not calculate the full set of order terms, but only some of these terms. Indeed, the notion of a systematic expansion in a small dimensionless parameter does not enter their calculational framework. As this example shows, convergent kinetic equations can be more accurate than the heuristic model building technique of the previous section, but one cannot be sure of their accuracy until a comparison with a systematic calculation has been made, as with BPS and Ref. (7).

Iii Bessel Function Example

iii.1 Analogy with Dimensional Continuation

I will illustrate the main points of dimensional continuation with an example involving the modified Bessel function , with an emphasis on analytic continuation and how this can be used to extract leading and next-to-leading order behavior. This example was first presented in Ref. (8), and for pedagogical purposes it was also included in Appendix A of Ref. (1). This example contains all the essential features of dimensional continuation, but in a mathematically simple form, and while it is an imperfect analogy, as all analogies are, it is explicit in all its details. We will show that the modified Bessel function has the expansion

(14)

to leading order (LO) and next-to-leading order (NLO) in , with denoting Euler’s constant. This expansion is quite accurate for small values , with an error of order rather than for symmetry reasons. The asymptotic expansion (14) is a well known result (9), but it is rather difficult to prove by conventional methods because of the non-analytic leading-log behavior. However, the method of dimensional continuation allows us to derive this result rather easily. The price one pays for this ease of derivation is that one must learn (or recall) a bit of mathematical machinery which, at first sight, seems unrelated to the problem at hand.

We start with the general integral representation of the modified Bessel functions (10),

(15)

As the notation in (15) suggests, we can think of as the dimension of space and the integration variable as the wave number. In this analogy, the argument corresponds to the dimensionless coupling parameter of the plasma. The following dictionary provides a useful mnemonic in relating this mathematical example to the plasma physics problem of real interest:

spatial dimension
wave number
(16)

Pushing our physical analogy further, if we think of as being the dimension of space, then it should always be a positive integer, which I will express by the conventional set theory notation ; however, nothing per se in the integral representation (15) requires that . We can therefore think of in expression (15) as being a continuous real variable (), or indeed, a complex variable () if circumstances warrant. Similarly, for a real physical system written in the appropriate integral form, there is nothing in any law of physics that prevents us from interpreting the dimension of space as being a complex number. Continuing from the positive integers into the complex plane is what I mean by “dimensional continuation.” There will be times when we restrict our attention to the real numbers only, rather than the complex numbers in general, and I will refer to this as dimensional continuation as well. As we shall see, this procedure of taking or will allow us to regulate otherwise infinite integrals in a systematic and perturbative fashion. Finite manipulations can then be performed, the divergent poles will cancel from physically measurable quantities, and afterward we can take to the appropriate integer dimension (in this analogy we take , rather than as we do for the physics problem).

iii.2 Leading Order Terms

Let us first calculate the leading order in behavior of for positive and negative values of . For small positive values of , the leading order -behavior can be obtained by replacing the exponential in (15) by one, except in the regions and , where the exponential is required for convergence. Taking first, note that the integral (15) is dominated by small values of near the lower limit of integration. In terms of the analogy (16) where is a wave number, this corresponds to the situation in which long-distance IR physics is dominant. Therefore, when and the integral is dominated by small values of , the leading order contribution to (15) can be obtained from the leading order behavior of the exponential, that is to say, the replacement

(17)

will capture the entire leading order in behavior for negative values of . Note that (17) provides convergence as , while large- convergence is provided by the prefactor since . We will denote this leading order contribution by , and using the substitution (17) we write

(18)

In the last equality of (18), we have made the variable change , and we have used the standard integral representation for the Gamma function,

(19)

This representation of the Gamma function only converges when (actually, (19) is valid over the complex -plane with ). When and , we may indeed use (19).

We can find the leading order in contribution when in a similar manner. In this case, the integral is dominated by large values of at the upper limit of integration, and we make the substitution

(20)

thereby giving the leading order result

(21)

Note that the exponential provides convergence as , while the integrand possesses an integrable singularity at for (the integrand is non-singular at when ).

We will eventually take the limits of (18) and (21), since we are interested in and not . While we could take to be a general real or complex number until the limit is required, it is easier to consider only small values of from the start (we need work no higher than linear order, since this and higher orders vanish when ). To do this, we expand the Gamma function to linear-order in its argument using . Taking in this expansion gives

(22)
(23)

Expressions (22) and (23) are accurate to linear order in the dimension ; on the other hand, (22) gives the leading order in contribution to as defined by (15) when , and (23) gives the leading order in contribution to when . To compare these two expressions, we must analytically continue them to a common dimension. We will discuss this further in the next section.

In terms of our physics analogy, expression (22) captures the leading order short-distance physics in the regime; the pole at corresponds to a small- divergence, which, pushing our physical analogy again, would reflect missing or incomplete long-distance physics (as with the Boltzmann equation). The situation is completely reversed for (23), which captures the leading order long-distance physics for , with the pole at corresponding to a large- divergence arising from missing short-distance physics (like the Lenard-Balescu equation). As functions of , we see from (22) and (23) that and are analytic in , except for the simple pole at . As we shall see, the analytic continuation to complex takes the same functional form as the individual expressions (22) and (23), each defined separately for and , respectively.

iii.3 Some Comments on Analytic Continuation

Since analytic continuation plays such a central role in dimensional continuation, at least mathematically, I would like to briefly discuss the conditions under which a function can be analytically continued from one region of the complex plane to another. Recall that a function is said to be analytic at a point in the complex plane , if and only if its derivative exists not only at , but also at every point within some open neighborhood of . A function is analytic on a domain in the complex plane if it is analytic at each point in . Analyticity is a very stringent condition on a function, since the existence of the derivative of a complex function is a much more robust constraint than the corresponding existence of the derivative of a function on a real domain. This is because in the two-dimensional complex plane, the limiting procedure defining the derivative must exist regardless of the direction used in taking the limit. In fact, analyticity at a point is so strong that it implies the existence and continuity of all derivatives for any order (11). In other words, an analytic function on can be thought of as being infinitely smooth on , even though the definition of analyticity itself invokes only the existence of the first derivative, albeit on a neighborhood.

Analyticity is such a stringent condition, that the behavior of an analytic function in a small domain is enough to determine its behavior in a larger region. Even if the function is only known along a one-dimensional curve in the complex plane, such as a portion of the real axis, this is enough to uniquely determine the function in the complex plane.9 This is because the derivatives of an analytic function exist to all orders, and therefore a Taylor series expansion about a point of analyticity always exists with some non-zero radius of convergence. For example, one of the powers of analytic continuation is that the original function may be defined in any manner over a subregion, and this can generate a unique function over a larger region. As an application of this, take the analytic function defined by the infinite geometric series

(24)

a series that converges only for . Upon defining by (24), we therefore take the domain to be the unit disk about the origin, excluding the unit circle itself. In this domain, the geometric series converges to

(25)

Note, however, that the function is defined over the entire complex plane except , a region I will call . Since the function is only defined within the unit circle, and since and agree within the unit circle, the function is the unique analytic continuation of .

As a more relevant example, consider with as given by (23). To compare this with , which is determined by (22) for , we must analytically continue to the positive real -axis. We can think of as a sequence of functions of an independent variable indexed by a continuous label ; therefore, in a more suggestive notation, we temporarily write  . While the collection of functions are defined in (23) on the negative -axis (excluding the simple pole at zero), they can be analytically continued to the complex -plane . Furthermore, the functions take the same algebraic form on the complex plane, namely,

(26)

We can now restrict our attention from in general to the positive -axis (excluding zero). This allows us to directly compare and at using the same algebraic forms as given by (23) and (22). In the next section, we discuss the implications of analytically continuing from the negative axis to the positive axis . Alternately, we could continue to the region using the same functional form as (22), and compare this with .

iii.4 Next-to-Leading Order Term

We now illustrate the key mathematical result that allows dimensional continuation to extract not only the leading, but the next-to-leading order terms. Recall from (22) and (23) that and are both leading order in for and , respectively. Since these functions were calculated for mutually exclusive values of , one might think that they cannot be compared. When viewed as an analytic function in the complex -plane, however, we have seen that is also a function over the domain , in which case both and can be compared at the same values of and . Since the algebraic form is so simple, takes the same functional form when analytically continued to as it did for . As illustrated in Fig. 1, this means that becomes next-to-leading order in along the positive real axis:

(27)
(28)

The expressions (27) and (28) remain accurate to linear-order in .

LO: large when NLO: small when analytically continuearound the pole
Figure 1: The analytic continuation of from to in the complex -plane: the same expression can be used for throughout the complex plane since the pole at can easily be avoided, as indicated in the figure. Note that is leading order in for . However, upon analytically continuing to , the -dependence becomes , and the function is next-to-leading order relative to  .

To see that (28) is indeed next-to-leading order in relative to (27), note that the -behavior of the leading order contribution for can be written . I have used the absolute value to emphasize that the power of in (27) is strictly negative when . Similarly, along the positive -axis we find the behavior for , and we see that for . This means for and , and we are therefore justified in calling leading order in and next-to-leading order.

Strictly speaking, we have only shown that is subleading relative to when . To conclude that is indeed next-to-leading order relative to , it is important to establish that there are no powers of between and in the expansion of . For , one simply subtracts (22) from (15), and it becomes clear that this error is higher order in than . For a more detailed proof of this, see footnote 2 of Ref. (8). A similar statement holds for , namely, as we analytically continue from to , the quantity switches from leading order to next-to-leading order in  relative to .

iii.5 Assembling the Pieces

We have now assembled enough results to find to leading and next-to-leading order in : we simply add the expressions (27) and (28) and take the limit of vanishing . Note that this does not lead to any form of “double counting.” Instead, we are simply adding the next-to-leading order term (28) to the leading order term (27) at a common value of . Upon taking the limit of vanishing , or more precisely since is always positive in (27) and (28), we obtain to leading and next-to-leading order in .

We now calculate this limit, proving that

(29)

Let us first expand in powers of . We will denote , from which we find , or in summary:

(30)

This allows us to express (27) and (28) as

(31)

and

(32)

When we divide (30) by a factor of , as required by (27) and (28), note that we find: (i) a pole from the first term in (30), (ii) a non-analytic finite contribution from the second term, and (iii) the error in becomes , which is the same order in that we are neglecting in (27) and (28). The error in , which vanishes in the limit , should not be confused with the error in , the latter being for vanishing . Note that the pole terms cancel upon adding (31) and (32), so that

(33)

thereby giving (29) as . As we have discussed, there are no -dependent terms that lie between and , so this procedure has captured the leading and next-to-leading order behavior in .

In exactly the same way, we can also calculate the leading order and next-to-leading order contribution to by taking the limit from the left,

(34)

I should point out a potential notational problem in (29) and (34). Concentrating on (29) for the moment, the limit indicates that both terms and are understood to live in dimensions , with the second term having been analytically continued from . The notation with which the term is written in (29), however, does not indicate that it has been analytically continued. This should be no cause for confusion, however, since takes the same functional form in any dimension ; therefore, a separate notation indicating that the in (29) has been analytically continued is unnecessary. We can simply add and as calculated in and respectively.

Iv Dimensional Continuation

iv.1 Rate of Energy Exchange as a Perturbative Expansion

The -Expansion

Before moving on to the details of dimensional continuation, we must first discuss the plasma expansion parameter . Since the problems in plasma physics are usually so complicated as to preclude a perturbative approach, most plasma physicists do not usually think in terms of expanding systematically in a small dimensionless parameter. However, for a weakly to moderately coupled plasma, it is a quite fruitful approach to perturbatively expand the rate in a small dimensionless coupling constant.

That such a universal parameter for a plasma exists was discussed at length in Ref. (13), where it was shown that any physical quantity associated with a plasma whose species are in equilibrium with themselves (such as the plasma we are studying) can be expanded in integer powers of a dimensionless coupling constant defined by

(35)

with being the Debye length of the plasma. Since the potential energy between two like charges is given by in rationalized units, and writing the Debye wave number as , the coupling parameter is therefore

(36)

For a multicomponent plasma, there is actually a coupling constant for each pair of components,

(37)

with defined by (4). However, when the pairs have approximately the same coupling strength, then the single parameter (36) adequately characterizes the entire plasma. Expressing the charges as and , we can write as

(38)

and we see that the coupling constant is proportional to the cube of the electric charge, the square root of the density, and the inverse (3/2)-power of the temperature.

Recall that the usual plasma parameter is defined in a similar manner to (35), except that the charge separation is determined not by , but by the inter-particle spacing in the plasma,

(39)

The inter-particle spacing is defined in several ways in the literature, but the idea is to transform the plasma number density into a length scale, so that . The most common convention is to define to be the radius of a sphere containing, on average, a single plasma particle, so that , and therefore

(40)

With this convention, the relation between the two plasma coupling parameters for a single plasma species is , and for an arbitrary number of plasma species we always find . We can therefore use either or to characterize the strength of the plasma, as and become large or small together.10 For our purposes, however, there is an an advantage to using rather than since physical quantities are expanded as integer powers of , while they expand in fractional powers of .

Next-to-Leading Order and the Coulomb Logarithm

As I have said, any plasma quantity can be written as a power series expansion in integer power of , with the possible exception of non-analytic terms involving . For the process of energy exchange via Coulomb interactions, this non-analyticity arises from the competition between disparate physical length scales. As an expansion in , the rate of energy exchange takes the form

(41)

In (41), I have indicated the leading order in (LO) and the next-to-leading order in (NLO) terms in the -expansion: the first term is leading order relative to the second because for small . The minus sign on the leading order term of (41) is a matter of convention. Since the logarithm will be negative in a weakly coupled plasma (recall ), the minus sign renders the coefficient positive when the energy exchange is positive. The coefficient was first calculated by Spitzer. The coefficient , however, is very difficult to calculate, and this was the main purpose of BPS (1). It is convenient to define the dimensionless coefficient by , in which case we can write

(42)

We see, then, that knowing the next-to-leading order term is equivalent to knowing the exact coefficient under the logarithm. Note that the minus sign renders the Coulomb logarithm positive when is very small, in keeping with convention.

Factors of Inside the Coulomb Logarithm

For the heuristic model building of Section II.1, let us pause for a moment and show that the argument of the Coulomb logarithm in (10) is indeed proportional to , as required by (42). On physical grounds we saw that the long-distance scale is set by a Debye length, and therefore we nominally set . In the extreme classical limit, the short-distance cutoff is set by the classical distance of closest approach , so that . For simplicity, we will choose the coefficient such that between (10) and (42), in which case

(43)

Let us consider two unit charges of mass approaching one another with zero impact parameter. The rms speed of each particle is determined by

(44)

while energy conservation gives the distance of closest approach,

(45)

In the extreme classical regime, we see that the argument of the Coulomb logarithm in (43) is indeed proportional to the plasma coupling constant,

(46)

Let us now look at the ad hoc interpolation (11) between the classical and quantum regimes. Up to this point I have said very little about quantum mechanics. While I will not dwell on quantum corrections, I will briefly discuss a dimensionless expansion parameter that characterizes the strength of the quantum two-body scattering correction. There are many ways of defining such a parameter, but I will follow Ref. (13), taking

(47)

With this definition, quantum corrections are large when . Motivated by the de Broglie wavelength of a particle, the thermal wavelength of a plasma species is given by , where is a typical momentum transfer suffered during a collision. Definition (47) yields , or more succinctly

(48)

from which (11) gives

(49)

In the extreme quantum limit in which , this becomes .

Finally, note that the factors inside the BPS Coulomb logarithm (3) are also proportional to the coupling constant , and upon dropping the electron subscripts for convenience, we find

(50)

We see that the -dependence of the Coulomb logarithm arises quite naturally. However, the accompanying coefficient under the logarithm might also possess -dependence, thereby obscuring the -dependence unless we are careful.

iv.2 Mathematics of Dimensional Continuation

Before describing what dimensional continuation is, allow me to first state what it is not. Dimensional continuation is not performing an integral to a fractional power of the spatial dimension, as with the meaningless expression

Instead, dimensional continuation is the following. Suppose some physical quantity of interest can be written as an integral over a kernel

(51)

where is determined by the physical equations of motion, whether classical or quantum. The integrand is of course a function of the physical parameters, such as the masses and charges of the fundamental particles, and I have abbreviated this dependence by the parameter . For definiteness, we will think of as a wave number with dimensions of an inverse length. The laws of physics, from which (51) follows, are usually written in three dimensional space. Thus, we usually take to be a three-dimensional vector, and we integrate over the entire three-dimensional Euclidean space .

The known fundamental laws of physics themselves, however, do not specify a particular spatial dimension in which they hold. In fact, as far as the known laws of physics are concerned, the actual value of the spatial dimension can be viewed as a free integer parameter: it is simply an unexplained empirical fact that we live in three dimensions.11 We can therefore express any three-dimensional physical quantity or law, such as Newton’s equation of motion or Gauss’ law, in any number of integer dimensions . For example, we can write down a corresponding quantity to (51) in an arbitrary number of dimensions,

(52)

where the wave vector is now a -dimensional vector, and we integrate over the entire -dimensional Euclidean space . I have placed a subscript on the integrand to indicate that it is determined by the theory expressed in dimensions. At this point, the spatial dimension is a non-negative integer, so that . Since the integer is arbitrary in (52), I have indicated that the corresponding quantity contains -dependence by writing ; however, for notational simplicity I will drop the parametric dependence of quantities such as mass and simply write . A quantity that diverges in three-dimensions will be finite in arbitrary , but it will typically exhibit a simple pole of the form

(53)

where is finite at .12 Dimensional continuation is simply the act of treating as a function of a complex argument , after the integral (52) has been performed for all positive integer values of . This is not an unfamiliar procedure, as the factorial function with can be generalized to the gamma function in which . Indeed, given any function defined on the integers, with only mild restrictions placed on the function at large values of the argument, Carlson’s Theorem (12) allows us to continue this function uniquely to the complex plane.

How does dimensional continuation work in practice? We will look at a few specific examples in Sections IV.3 and VI, but for now let us consider a general physical quantity in which the integrand in (51) depends solely upon the modulus of , so that

(54)

In such a case it is not uncommon that the integrand in the generalization (52) is only a function of the modulus of the -dimensional wave vector , with the same functional form as the integrand in (54). In other words, in (52) we have with , thereby allowing us to write

(55)

Since the integrand is a function only of , we can extract the angular integrals and write

(56)

At this point, the dimension is simply an arbitrary positive integer, . As we will show in the next paragraph, the integration over all angles gives

(57)

This leaves a one-dimensional integral to perform, in which simply acts as a parameter,

(58)

The physical quantity now becomes the product of (57) and (58) with . In the case of (57), we already know how to analytically continue to complex values. On the other hand, for (58) we can think of as being an arbitrary complex number when performing the one-dimensional integral over , and therefore we can regard as a function over the complex -plane, thereby giving

(59)

In this manner, we can regard as a function of a complex argument , and by Carlson’s Theorem (12), this is the unique continuation from positive integer values of to complex values of .

As an example of this procedure, let us prove (57). First, consider the one-dimensional Gaussian integral

(60)

If we multiply both sides together times (with ), we find

(61)

where the wave vector in the exponential of the last expression is the -dimensional vector , and