Appearance and disappearance of thermal renormalons
We consider a scalar field model with a interaction and compute the mass correction at next-to-leading order in a large- expansion to study the summability of the perturbative series. It is already known that at zero temperature this model has a singularity in the Borel plane (a “renormalon”). We find that a small increase in temperature adds two countable sets both with an infinite number of renormalons. For one of the sets the position of the poles is thermal independent and the residue is thermal dependent. In the other one both the position of poles and the residues are thermal dependent. If we consider the model at extremely high temperatures, however, one observes that all the renormalons disappear and the model becomes Borel summable.
The understanding of strongly-coupled systems remains one of the major challenges in particle physics and requires the knowledge of the nonperturbative regime of Quantum Chromodynamics (QCD), the currently accepted theory of strong interactions.
Also, in the realm of condensed matter physics, systems involving strongly coupled particles (fermions, for instance) fall, in principle, outside the scope of perturbation theory. However, apart from some simple models, nonperturbative solutions are very hard to be found, which led along the years to attempts to rely in some way on perturbative methods (valid in general for weak couplings) to get some results in strong-coupling regimes Le Guillou and Zinn-Justin (2013); Aniceto and Schiappa (2015); Grassi et al. (2015); Cherman et al. (2015a); Dorigoni (); Bender and Orszag (1999).
It is broadly discussed in the literature whether nonperturbative solutions in field theory can or cannot be recovered from a perturbative expansion. In any case, a procedure is needed to make sense out of the perturbative series. In fact, often the perturbative expansions are asymptotic rather than convergent. Actually, we remember that the perturbative series can be viewed just as a representation of the exact solution and if we want to obtain information about the nonperturbative solution from its perturbative representation some summation technique must be implemented. Le Guillou and Zinn-Justin (2013); Aniceto and Schiappa (2015); Grassi et al. (2015); Cherman et al. (2015a); Dorigoni (); Bender and Orszag (1999)
One of the most employed of these procedures is to investigate, after perturbative renormalization has been performed, the so-called Borel summability of a theory, for a brief introduction see Refs. Rivasseau (1991); Khanna et al. (2009) and for a complete review on the subject see Ref. Hirsbrunner (1981) . If we start with an asymptotic series, its Borel transform defines a new series that can be convergent. The representation of the nonperturbative result can be obtained by an inverse Borel transform, essentially a Laplace transform, which requires a contour integration in the complex Borel plane in order to be properly defined. This procedure allows one to gain access to the correct nonperturbative solution in many situations Grassi et al. (2015). More precisely: if we take a theory characterized by an already renormalized coupling constant and consider a given quantity given by a formal series (perhaps asymptotically divergent) in , ; define its Borel transform as and the inverse Borel transform as . It can be easily verified that reproduces formally the original series . The interesting point is that even if is divergent the series may converge and in this case the inverse Borel transform defines a function of which can be considered in some sense as the sum of the original divergent series . This “mathematical phenomenon” is named Borel summability and is a way of giving a meaning to divergent perturbative series. However, it is implicitly assumed the absence of singularities (renormalons) at least on the real axis of the Borel plane .
On the other hand, the existence of poles in the complex Borel plane makes the procedure ambiguous and thus ill defined. These difficulties are essentially of a nonperturbative nature. Recent developments have investigated this issue considering models for which exact results are known, so that we can be sure that the inverse Borel transform or other techniques can give information about the nonperturbative behavior Grassi et al. (2015); Cherman et al. (2015b). This quest introduces the resurgence technique, developed by Écalle in a different context J. Écalle (1981) which has led to many applications Shifman (2015); Beneke (); Cherman et al. (2015a, b); Misumi et al. (2015); Aniceto et al. (2015); Dunne and Ünsal (2012); for a review see Refs. Dorigoni (); Aniceto et al. (); Mitschi and Sauzin ().
As pointed out in Shifman’s review article Shifman (2015), the study of renormalons is also important from a phenomenological perspective, as one needs, for instance, to know the solution at the nonperturbative level to obtain an estimate of the heavy-quark mass Beneke and Braun (1994).
Recently, the investigation of renormalons and the subsequent task of the resurgence program to “cure” the theories have been done, considering compactified theories such as non-Abelian gauge theories on Anber and Sulejmanpasic (2015) and the non-linear sigma model on Cherman et al. (2015a); Shifman and Ünsal (2008). It seems that the renormalon ambiguities are canceled by appropriate contributions from instanton-anti-instanton pairs in these theories Cherman et al. (2015a). The interest of considering a finite spatial extent or thermal dependence arises from the fact that for small or large a weakly coupled regime is observed due to asymptotic freedom Malbouisson et al. (2004); Khanna et al. (2012).
A careful study of renormalons for an gauge theory has been made in Ref. Anber and Sulejmanpasic (2015). In it the absence of renormalons is discussed when one introduces a finite small length . Another proposal Shifman (2015) is to study 2D models with symmetries such as , , where the small-length regime becomes a quantum mechanical problem.
In the present article, we investigate the behavior of a scalar field model with symmetry in four dimensions. Our main concern is the careful investigation of the renormalon poles and residues at next-to-leading order in the large- expansion. This -expansion allows to resum a class of diagrams (usually called ring diagrams or necklaces) that generates the renormalon contribution. Following recent literature, we investigate the role of a compactification parameter, here taken as introducing a temperature dependence. First, we review the behavior at zero temperature and find the existence of two renormalons. At small temperatures we observe that the system develops a countable set with an infinite number of renormalons that can be separated into two classes: renormalons without thermal poles but that can have thermal residues and renormalons with thermal poles. A further increase in temperature is found to imply the disappearance of renormalons.
Ii Scalar field model and resummation
We are mainly interested in computing corrections to the field mass in a scalar theory with coupling . The full propagator is given by
where is the sum of all 1PI (one-particle irreducible) diagrams built with the free propagator . Or, if we establish using the full propagator (a recurrence relation) then, to avoid double counting, it is necessary to consider just the 2PI diagrams. We use a set of 2PI diagrams known as necklace or ring diagrams as ilustrated in figure 1.
Therefore, at an unspecified spacetime dimension a necklace with -pearls is given by ,
Thus, taking into account necklaces with all numbers of pearls we obtain the full correction
The subsequent analysis of this expression intends to verify whether the series representation is or is not Borel summable. This is entirely dependent on the behavior of with respect to the summation index .
Now that we have introduced the general idea, let us investigate the thermal dependence in detail. By making a compactification in imaginary time we introduce the inverse temperature . With this, the expression for each pearl is modified to
where is the frequency related to the -dimensional momentum , while the necklaces become
where and are the frequencies related, respectively, to the loop momentum and the external momentum .
Then, Eq. (5) can be treated by using the Feynman parametrization, integrating over the momenta and identifying the infinite sum as an Epstein-Hurwitz zeta function defined by
where is the modified Bessel function of the second kind.
Considering the case where , we get
The temperature-independent component is standard and well known Zinn-Justin (2002),
For high values of the momentum we have the asymptotic expression
Iii A first glance : renormalon at
In this section we consider the special case of zero temperature. In this situation the only contribution to the pearl diagram comes from the component. To obtain a treatable expression to the necklace diagrams, we consider the expansion for high values of the momentum , Eq. (11), at zero temperature
At this point we recall that the standard approximation is to consider the leading behavior in the momentum , that is, . Here, we avoid this particular approximation and explore the consequences of keeping the exact term . Let us return to the necklace diagrams. We employ a BPHZ procedure defining a renormalized necklace
We shall drop the hat unless it becomes important to distinguish between the renormalized and non renormalized necklaces.
As can be noted, this affects only the -dependent propagator in the zero-temperature version of Eq. (6). The procedure is equivalent to perform the substitution
where the standard naive expansion is assumed.
In a low- expansion, then, for and small values of , the integral to be solved to obtain the necklace expression is
To solve it we first perform the integral over the solid angle () and then reorganize the result by making the change of variables , that is,
At this point we can clearly identify the presence of two gamma functions, so that
We can finally return to the sum over all contributions Eq. (4),
where we have defined . Both sums are divergent, but we can try to make sense of them by defining a Borel transform,
We then obtain two poles on the real positive axis of the Borel plane at . These poles (renormalons) introduce problems to compute the inverse Borel transform.
In the standard procedure, see Ref. Zinn-Justin (2002), Eq. (15) is solved for very large , which is justified as this is the relevant region to get the asymptotic behavior for the index. This means that the approximation is employed. Therefore, only the first pole is found (at ) while the second pole is hidden. When is very small this could be justified as being very far from the origin.
Iv Appearance of thermal renormalons (low temperatures)
For low but finite temperatures, we can use the asymptotic representation of the modified Bessel function of the second kind , so that the thermal component of the pearl (9) becomes
Using the above equation for and the expression for the component, (see Eq. (11)), the quantity can be written in the low-temperature regime as
stores information about the dependence on the temperature.
So, integrating over the solid angle and expanding the binomial, we get
We reorganize the above expression in a more convenient way to compute the sum over the Matsubara frequencies. The denominator is treated by employing a Feynman parametrization and the logarithm in the numerator is expanded in powers of , which is justified by an asymptotic behavior in assuring that . This allows to rewrite the above equation in the form
Although we could use this complete expression, this is unnecessary. It can be shown, after a lengthy computation, that the relevant information (poles in the Borel plane) can already be obtained by using the following approximation,
where is the Epstein-Hurwitz zeta function defined in Eq. (7). The contributions of order do not modify the position of the poles and only change their residues. Moreover, for large values of the integration of the expression over the Feynman parameter is asymptotically equal to the expression without the Feynman parameters. To avoid a tedious calculation we do not exhibit in this article the step-by-step of this process.
Taking the approximation in Eq. (26) and considering again the analytic expansion of the Epstein-Hurwitz zeta function to the whole complex plane, we get
Since is an integer, the modified Bessel function of the second kind has a half-integer order, which has the series representation Gradshteyn and Ryzhik (2007)
So, the remaining integrals are given by
The first integral in the preceding equation can be solved as in the zero-temperature case (see Sec. III) by the change of variables . One must note that has an upper bound that is also its asymptotic value for large values of the momentum (which means also large values of the index ). Then, we can use that to simplify the integral
For the second integral in Eq. (29) we make the change of variables so that we obtain
Once more, we used that , which considerably simplify the integral and allows it to be identified as the Milgram generalization of the integro-exponential function whose asymptotic behavior is known Milgram (1985)
Now, let us focus attention on the -dependence. The previous equation can then be rewritten as
where we have defined,
The sum over all necklaces is then
The range of summation for the double sum is ; we can change the sum ordering and then split the sum over the index in the form
The first double sum has the dominant contribution; this can be seen by checking for each value of . Therefore, the relevant contribution is
However, this is not summable due to the presence of the . To overcome this difficulty we can employ a Borel transform,
Finally, we see in Eq. (37) the renormalons that appear for low temperatures. There are two different sets of renormalons both with residues that are thermal dependent, respectively and . The first set of renormalons was already found in previous works Loewe and Valenzuela (2000); it is characterized by poles whose position are thermal-independent and they are located along the real axis at positions for . However, the second set of poles, as far as we know, has not yet been reported. These poles are also in the real axis but they are thermal-dependent as they are located at for . The existence of this new set seems to be a remarkable enrichment for the model.
We remark that in the limit of extremely small temperatures this new set of renormalons are all very far from the origin and this may justify why they are usually hidden. Therefore, our result can be viewed as a first correction to the standard approach. Furthermore, as we pointed out before, we claim that our approximation in Eq. (26) is the sufficient one (at least to describe the poles) and any further corrections shall only change the residues. This means that we have mapped all the renormalons that appear at low temperatures.
As a further comment, we remember that in Sec. III we show that at zero temperature there is a hidden second pole located at . This does not add any new poles at low temperatures because, as can be easily noted, we already have an infinite set of poles located at for .
V Disappearance of thermal renormalons (high temperatures)
In this section we explore the regime of very high temperatures. In this situation, to treat Eq. (9) we can use the following series expansion of the modified Bessel function of the second kind Gradshteyn and Ryzhik (2007),
The result is easier to get by assuming from the beginning that (which means that this is the only relevant mode) and recalling the following properties of the Riemann zeta function, ,
Remembering Eq. (9), we then obtain the result
revealing that at high temperatures the original contribution from zero temperature is not present anymore. This has a major impact and is responsible for the disappearance of the renormalons. Therefore, we may write
If we replace this back into the necklace expression , in Eq. (23), we get
Since the integration over the internal loop is independent of