TUMHEP870/12
TTK1249
December 12, 2012
On “dynamical mass” generation in Euclidean de Sitter space
M. Beneke and P. Moch
[5mm] Physik Department T31,
Technische Universität München,
JamesFranckStraße 1, D  85748 Garching, Germany
[0.3cm] Institut für Theoretische Teilchenphysik und Kosmologie,
RWTH Aachen University,
D–52056 Aachen, Germany
[0.5cm]
Abstract
We consider the perturbative treatment of the minimally coupled, massless, selfinteracting scalar field in Euclidean de Sitter space. Generalizing work of Rajaraman, we obtain the dynamical mass of the scalar for nonvanishing Lagrangian masses and the first perturbative quantum correction in the massless case. We develop the rules of a systematic perturbative expansion, which treats the zeromode nonperturbatively, and goes in powers of . The infrared divergences are selfregulated by the zeromode dynamics. Thus, in Euclidean de Sitter space the interacting, massless scalar field is just as welldefined as the massive field. We then show that the dynamical mass can be recovered from the diagrammatic expansion of the selfenergy and a consistent solution of the SchwingerDyson equation, but requires the summation of a divergent series of loop diagrams of arbitrarily high order. Finally, we note that the value of the longwavelength mode twopoint function in Euclidean de Sitter space agrees at leading order with the stochastic treatment in Lorentzian de Sitter space, in any number of dimensions.
1 Introduction
It is wellknown that the free, massless, minimally coupled scalar field in the de Sitter background spacetime cannot be defined in a de Sitter invariant way due to infrared (IR) divergences [1]. This occurs in any number of dimensions , since the mode integral that defines the Wightman twopoint function always diverges logarithmically for . In the expanding cosmological coordinate frame the divergence arises from the redshifting of modes, which leads to a pileup of longdistance modes at late times. But noninteracting fields are not very interesting. The question arises whether upon turning on an interaction of the scalar quantum field, no matter how small, the IR problem could somehow cure itself. For this to happen, the interaction must effectively become nonperturbatively strong among the longdistance modes. If so, the nonperturbative dynamics may be too complicated to be solved with analytic methods. However, it may also be that after a suitable resummation or reorganization of the expansion in the interaction strength, the interacting, massless, minimally coupled scalar field lends itself to a welldefined, systematic treatment.
Various previous results suggest that in scalar field theory with a quartic selfinteraction the originally massless field acquires a dynamical mass , where is the Hubble constant of de Sitter spacetime, which indeed regularizes the IR divergence. Starobinsky and Yokoyama [2] treat the longdistance fluctuations of the field as a classical random field that satisfies a Langevin equation. The associated FokkerPlanck equation is solved for large times by a probability distribution that results in finite correlation functions. Another approach uses the SchwingerDyson equations and obtains the dynamical mass from a selfconsistent solution. In the meanfield or the large limit [3, 4] the selfenergy can be restricted to the oneloop, tadpole diagram. Garbrecht and Rigopoulos [5] analyzed the various inin propagators in the CTP formalism and found that the large result is modified by the twoloop selfenergy, but remarkably, no further contribution arises beyond two loops due to systematic cancellations in the CTP index sums. However, while both formalisms agree on the parametric size of the dynamical mass squared, the two exact results from [2, 5] disagree on the numerical prefactor. Neither of the formalisms so far explains how to compute subleading terms systematically.
The present work is motivated by the attempt to resolve the difference between the classical stochastic and diagrammatic/SchwingerDyson approach. For reasons that will become evident it is much simpler but still instructive to investigate the issue in Euclidean de Sitter space, which is simply the sphere . In an elegant paper Rajaraman [6] considered the functional integral on the sphere and identified the zeromode integral as the origin of nonperturbative dynamics. He computed the twopoint function of the zero mode, which can be related to the dynamical mass. In this paper we extend the functionalintegral approach and use 2PI methods to compute the exact selfenergy, which is the central quantity in the diagrammatic approach. Our main results are as follows:

We formulate the rules for a welldefined perturbation expansion of correlation functions of the massless, minimally coupled scalar field in Euclidean de Sitter space. The expansion parameter turns out to be instead of the coupling of the standard perturbation expansion.

For the massive scalar field we obtain the dependence of the dynamical mass on the Lagrangian mass ; for the massless field the leading correction.

We show that the dynamical mass can be obtained from the loop expansion of the selfenergy after summing a divergent series to all orders in the loop expansion.

It seems to have gone unnoticed that the Euclidean dynamical mass [6] agrees with the result from the stochastic approach. We show that this is true in an arbitrary number of spacetime dimensions despite the fact that the relevant dimensiondependent quantities are apparently unrelated.
The interacting, massless, minimally coupled scalar field is therefore perfectly welldefined on the de Sitter background. For there is a systematic weakcoupling expansion. The reason why this is possible despite the fact that the zero mode is truly strongly coupled is that the infrared theory consists of a single degree of freedom (the zero mode), whose dynamics can be solved exactly. What all this implies for Lorentzian de Sitter space is less clear. We must leave this important point to further investigation.
2 Scalar field in Euclidean de Sitter space
Euclidean de Sitter space is obtained from dimensional de Sitter space in global coordinates with line element
(1) 
by defining and assuming periodicity in , which turns into
(2) 
Thus, Euclidean de Sitter space is equivalent to the dimensional sphere with radius . Because the sphere is compact, functions admit a discrete mode expansion in spherical harmonics. In dimensions the spherical harmonics are labelled by the integer index vector with and satisfy (see, e.g., [7])
(3) 
as well as the orthogonality relation
(4) 
The volume of Euclidean de Sitter space is
(5) 
Here denotes the lowest harmonic, which is constant.
We consider the minimally coupled, real scalar field with Euclidean action
(6)  
where the second line follows from the mode expansion
(7) 
and the orthogonality relation (4). From the quadratic terms of (6) we deduce the free propagator
(8)  
with
(9) 
Here is the invariant distance on the sphere , and , are two unit vectors on the subsphere with solid angle element in (2). The second line of (8) is indeed the de Sitter propagator in the BunchDavies vacuum [8, 9] in imaginary time.
The free propagator is illdefined for . The leading term for small is
(10) 
which, as can be seen from the first line of (8), originates only from the zero mode. Let us separate the constant zero mode from the field by defining
(11) 
The free propagator is the sum of the zero mode and nonzero mode propagator, since cross terms vanish by angular momentum conservation. The free zeromode propagator equals the righthand side of (10) for any / , while the nonzeromode propagator has a welldefined massless limit. In ,
(12) 
From now on we consider the massless, scalar field, , unless mentioned otherwise, and assume . The free zeromode propagator is not defined, which is not surprising, since the zero mode has no quadratic term in the action (6). The zeromode must be treated nonperturbatively. Let
(13) 
be the exact twopoint function of the interacting theory. Since is constant, we may write
(14) 
Comparison with the term in the first line of (8) suggests that we identify with the dynamical mass of the originally massless scalar field, generated by the selfinteraction. Note that this interpretation should be regarded with some caution, since the value of is not related to the decrease of correlation functions at large separation . In fact, corresponds to distances parametrically larger than the radius of the sphere, which carry no meaning. Similarly, in Lorentzian de Sitter space a dynamical mass of order is related to superhorizon correlations. Nevertheless, as will be seen below, a finite value of regularizes the IR divergence of the massless field and allows us to define a wellbehaved perturbation expansion.
3 Perturbation expansion on the sphere
In [6] the zeromode twopoint function was computed by evaluating the dominant contribution to its functionalintegral representation, which gives
(15) 
In the following we generalize this approach. We show that both, and have welldefined perturbation expansions in , and provide a set of Feynman rules for this expansion.
The generating functional is conveniently written in terms of two separate sources , , for the zero and nonzeromode field, respectively:
(16)  
where is defined such that . Here
(17) 
and the term proportional to vanishes since for . The key point is that the term is not included in , but must be part of , since in the absence of a mass term for the scalar field the quadratic term in the zeromode action vanishes, and the integral over in does not converge for large field values [6]. Hence,
(18) 
while the generating functional for the free nonzeromode field,
(19)  
is a standard Gaussian functional integral. The zeromode functional integral is simply an ordinary onedimensional integral. Moreover, and are independent of , so in (18). The integral over can be evaluated exactly. Introducing
(20) 
we find
(21)  
where denotes a hypergeometric function. One easily checks that
(22) 
reproduces (15) as is should be. Similarly, follows from taking the appropriate number of derivatives. The index 0 on the bracket means that the computation is done with the zeromode functional alone. The full zeromode point functions computed from in (16) receive subleading corrections, as discussed below.
It is now straightforward to develop a systematic perturbative expansion of (16) and the corresponding Feynman rules. From (18) it follows that every zeromode fields counts as , while has the standard counting 1. The interaction therefore counts as . In general, since there is always an even number of involved, correlation functions have an expansion in . The rules are as follows: For a given correlation function expand (16) to the desired order in using the above counting rules. Perform the standard Wick contractions of pairs of nonzero mode fields. This can be represented in terms of lines and vertices in the usual way. However, no Wick contractions are to be performed for the zeromode fields. Instead, collect all factors of and compute the expectation value of exactly.
As an example, we evaluate the first correction to the zeromode and nonzeromode twopoint functions. For the zeromode case, we have
(23)  
where the black square represents the vertex. If we define as before in (14) and denote the previously obtained leadingorder expression (15) by , the previous equation translates into
(24) 
where has been used. While the leading expression is unambiguous, the first correction depends on the UV subtraction that defines the coincident nonzeromode propagator . To be specific, consider the case of four dimensions. In dimensional regularization one needs to take before expanding the dimensional propagator around , in which case
(25) 
The renormalized value corresponds to this expression with the pole term in in brackets subtracted. Further, is the renormalization scale. We note that the dynamical mass is not by itself a physical quantity. While the leading term is unambiguous, the first correction involving the propagation of nonzero modes is scheme and scaledependent.
The nonzeromode twopoint function is the free propagator in leading order. Including the correction it reads
(26)  
The expression for the leading correction differs from [6]. The diagram computed there is part of the subleading correction. In four dimensions the leading term equals (12), and the leading correction can also be summed to give
(27) 
where .
4 SchwingerDyson equation
We now return to the approaches pursued in [3, 5, 4] which are based on evaluations of the scalarfield selfenergy and the SchwingerDyson equation
(28) 
We project this equation on the zeromode component by integrating over and using the identity
(29) 
which follows from (4). This results in
(30) 
In the spirit of the 2PI formalism (see below) we regard the selfenergy as a functional of the exact propagator and derive it from the functional derivative
(31) 
of the 2PI effective action with the classical and oneloop term subtracted (as denoted by “rest”) [10]. The loop expansion of the effective action is given by
(32)  
where denotes the combinatorial factor associated with a diagram, as given in the first line, and where we have given the explicit diagrammatic representation up to the fiveloop order. Instead of appealing to the 2PI formalism we could have written down the selfenergy diagrams directly with the proviso that all internal lines are exact rather than free propagators.
The powercounting rules of the previous section tell us that the leading contribution to and is obtained from pure zeromode diagrams, that is, every full propagator is replaced by . Since every loop brings one factor of from the new vertex and adds two propagators, which each count as , we conclude that every order in the loop expansion contributes to the leading term. This shows that the loop expansion must be summed to all orders to obtain the correct value of the “dynamical mass” in Euclidean de Sitter space.
To see this explicitly, note that upon plugging the derivative of
(32) into (30), the latter equation can
be solved for , which yields the value of
.
(More precisely,
since by replacing by in (32)
we pick up the leading term only.)
The functional derivative in (31) eliminates two
integrations in (32) such that an loop selfenergy
diagram contributes
to (30).
(33) 
where the term corresponds to the sum of loop selfenergy diagrams. In terms of , the “dynamical mass” is given by
(34) 
Keeping only the oneloop tadpole diagram in (33), which corresponds to truncation after the quadratic term, we obtain and (in ), which coincides with the meanfield result [2] and the oneloop result in Lorentzian de Sitter space [3, 5, 4]. In the twoloop approximation, the quadratic equation for does not yield a real solution for . Thus, at two loops there is a difference between the solution of the SchwingerDyson equation in Euclidean de Sitter space and the solution to the corresponding equations for the closedtimepath propagators in Lorentzian de Sitter space [5]. Continuing to higher orders in (33), at three loops, we find , while at four loops (which includes the last term shown explicitly in (33)) there is again no solution. Thus, the loop expansion does not seem to converge to the exact value (15), which corresponds to
(35) 
The question arises how the exact result that is obtained easily from the functional integral is recovered diagrammatically. Clearly, we need the expansion of to all orders. But this cannot be obtained from (32). While the integrations are trivial in the zeromode approximation, the diagram topologies and computation of combinatorial factors become too complicated.
5 Zeromode dynamics in the 2PI formalism
In the following we exploit the 2PI formalism [10, 11] to derive an expression that generates the perturbative expansion of the zeromode selfenergy to any desired order. We focus on the zeromode dynamics which alone is responsible for the leading contributions as discussed above, and hence set to zero. In this section we drop the subscript “0”, since all quantities are understood to refer to the zero mode.
The generating “functional” in the 2PI formalism is
(36)  
where and (equal to ) have been defined in (20), , and
(37) 
The simple onedimensional integral in the second line of (36) applies since the zeromode field is constant. The exact propagator in the presence of the external sources can be found from the relation
(38) 
where is the field expectation value. Since the functional integral is an ordinary onedimensional integral, the functional derivatives are actually ordinary derivatives. It follows from (21) that the field expectation value vanishes in the absence of the source , that is, the symmetry is not spontaneously broken. This remains true for and . Since eventually we are interested in the theory in the absence of external sources we now put and consequently . In this case we find the closed expression
(39) 
where denotes the modified Bessel function of the second kind. From (38) we obtain
(40) 
The previous equation gives the exact zeromode propagator or, equivalently, the “dynamical mass” of the zero mode in the presence of the propagator source . We note that since is constant, has an equivalent interpretation as a Lagrangian mass for the scalar field. Hence (40) provides the “dynamical mass” of the scalar field for arbitrary , generalizing the expression (15) for the massless case [6]. The dependence of on is sketched in Figure 1. At large the “dynamical mass” asymptotes to with ordinary perturbative corrections of order , as should be expected, since the infrared enhancement that renders the zeromode dynamics nonperturbative is cut off for a sufficiently massive scalar field. For , the “dynamical mass” tends to the value (15). The preasymptotic corrections can be determined easily by expanding (40) around the corresponding limits.
6 Diagrammatic zeromode selfenergy to all orders
We now determine the selfenergy that is needed to solve the SchwingerDyson equation. In the 2PI formalism, the SchwingerDyson equation reads
(41) 
The inverse free propagator is , which vanishes for . This reflects once more the fact that the free propagator of the massless scalar field is illdefined in the absence of external sources. Thus,
(42) 
It follows that the diagrammatic expansion of the selfenergy is obtained by inverting given in (40), and expanding it in powers of .
While a closed expression for the inverse of may not exist, we can solve for the expansion in by making the ansatz
(43) 
where the first term is required by (42) to obtain a regular perturbative expansion of . The term represents the sum of the loop diagrams to the zeromode selfenergy, expressed in terms of the exact zeromode propagator. The definition of the expansion coefficients is chosen such that with the definition (34) of the function defined in (33) is given by
(44) 
Plugging the ansatz (43) into (40) and matching coefficients in the expansion in (equivalently, in ), we find the . The first ten terms are shown in Table 1. We note that the first four agree with (33) obtained from the combinatorial factors of the lowestorder Feynmandiagram topologies. We determined the exact coefficients up to , which turn out to be rational numbers of increasing length.
1  2  3  4  5  6  7  8  9  10  

Inspection of the coefficients shows that they form a signalternating, factorially divergent series with
(45) 
The divergent behaviour arises because the expansion in small corresponds to an expansion of around , see (43), while the value of is related to at . The divergent series is also the reason why we did not obtain a reasonable approximation to from the loworder approximations to the selfenergy.
It remains to show that the SchwingerDyson approach is consistent with the exact result (15) for the “dynamical mass”, which requires summing the divergent series. To this end we construct the Borel transform of
(46) 
such that the Borel sum of is given by
(47) 
Since the series is signalternating, we expect to exhibit a singularity on the negative axis, but without a closed expression we do not know the precise singularity structure of the Borel transform of . Given (40), it is reasonable to assume that it is analytic in a vicinity of the positive real axis such that the Borel integral is welldefined, and to assume that the Borel sum equals the original function .
Termbyterm integration of the series expansion of simply returns the divergent series expansion of . We therefore resort to a standard trick [12] and construct a Padé approximation from the truncated series expansion. More precisely, we use the first coefficients of the expansion of , not counting the “1” in (46), and construct the diagonal Padé approximant. We use this approximation to in the Borel integral (47) and obtain by numerical integration. We then solve the equation , see (33), to determine the “dynamical mass”. Alternatively, we can determine the solution of from Padé approximants to the expansion (44) of directly, without going through the Borel transform. The results are shown in Table 2. The solutions are seen to quickly approach the exact result (35), especially when the Padé approximation is applied to the Borel transform, in which case onepermille accuracy is reached already for . This demonstrates that the diagrammatic approach via the 2PI SchwingerDyson equation reproduces the pathintegral result, as it must be, but only after summation of a divergent series expansion to all orders.
3  6  9  12  24  48  96  

from  1.65709  1.65635  1.65581  1.65580  1.65580  1.65580  1.65580 
from  1.71012  1.66262  1.65723  1.65618  1.65581  1.65580  1.65580 
7 Stochastic approach in dimensions
The methods applied above do not extend to Lorentzian de Sitter space, which is noncompact, and does not allow to identify the (leading) infrared dynamics with the one of a single zeromode degree of freedom. However, quite some time ago Starobinsky and Yokoyama [2] suggested that the longwavelength part of the scalar field can be treated as a classical stochastic variable, which satisfies a Langevin equation with a random force provided by the shortwavelength modes. Here we show that this leads to the same value for the twopoint function of the longwavelength field as the zeromode twopoint function in Euclidean de Sitter space, in any number of dimensions . This intriguing coincidence seems not to have been noted before.
Following [2] we divide the scalar field into , where contains all longwavelength modes with wave number with the scale factor and the parameter that separates long from short wavelengths. From the field equation it follows that satisfies the Langevin equation
(48) 
Here is the scalar field potential and the stochastic force
(49) 
generated by the shortdistance modes. At leading order, we can neglect the selfinteraction of the shortdistance modes. The fluctuations satisfy with
(50) 
The first factor arises from the volume of the dimensional momentum shell , the second from the longwavelength limit (since ) of the BunchDavies mode functions . The FokkerPlanck equation for the oneparticle probability density associated with (48) is
(51) 
which admits the stationary latetime solution
(52) 
in terms of which the twopoint function of the (constant) longwavelength field is given by
(53) 
This precisely agrees with (22) (for ) provided the dissipation and fluctuation coefficients in the FokkerPlanck equation are related to the volume of dimensional Euclidean de Sitter space with radius by
(54) 
which can be easily verified. Hence, the longwavelength twopoint functions (and therefore “dynamical masses”) are the same, as claimed. One may wonder why the result could be derived from zeromode dynamics alone in Euclidean de Sitter space, while the fluctuations originated from the shortwavelength modes in the stochastic approach. However, the result from the latter is independent of in the above approximation, as it should be, and the stochastic force is generated by wave numbers . We can take arbitrarily small and conclude that in the leading approximation the main contribution to the stochastic force can be assumed to originate from the boundary between long and short wavelengths, which can be taken to be deep in the infrared.
We note that the stochastic approach can be derived rigorously from the full quantum dynamics in the leading logarithmic approximation in [13], which is equivalent to keeping the leading infraredenhanced terms in Euclidean de Sitter space. But unlike the Euclidean case discussed in the present paper, a systematic method for calculating corrections around the Lorentzian result of [2] is not known.
8 Conclusion
In this paper we considered the perturbative treatment of the minimally coupled, massless, selfinteracting scalar field in Euclidean de Sitter space. Generalizing the work of Rajaraman [6], we obtained the dynamical mass of the scalar for nonvanishing Lagrangian masses, as well as the first perturbative quantum correction in the massless case, and developed the rules of a systematic perturbative expansion, which after treating the zeromode nonperturbatively, goes in powers of . We then showed how the dynamical mass can be recovered from the summation of the diagrammatic expansion of the selfenergy and a consistent solution of the SchwingerDyson equation. This clarifies the relation between the pathintegral and diagrammatic treatment, and implies that solutions based on truncations of the loop expansion can at best be approximate. With the proper exact treatment of the zero mode, the reorganized perturbative expansion is free from infrared divergences, which are present for the free, minimally coupled scalar field in Euclidean de Sitter space. The interacting, massless field is therefore welldefined, and the rules for generating the systematic perturbative expansion are almost as simple as the standard rules for the massive case.
What this implies for Lorentzian de Sitter space is much less clear. We showed that the longwavelength mode twopoint function computed in the stochastic approach of [2] coincides with the exact Euclidean result in leading order in the expansion in . This strongly suggests to us that the dynamical mass of the selfinteracting scalar field in de Sitter space can be obtained by some sort of analytic continuation from the Euclidean, up to higherorder corrections. It would be very interesting to derive this result diagrammatically, in the spirit of [5], and to understand how to develop a systematic expansion in Lorentzian de Sitter space.
Acknowledgement
We thank B. Garbrecht, T. Prokopec and G. Rigopoulos for discussions. This work is supported in part by the Gottfried Wilhelm Leibniz programme of the Deutsche Forschungsgemeinschaft (DFG).
Footnotes
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