Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model
Valentin Bonzom, Etera R. Livine, Matteo Smerlak, Simone Speziale
aa r X i v : . [ g r- q c ] J a n Towards the graviton from spinfoams:the complete perturbative expansion of the 3d toy model
Valentin Bonzom,
1, 2, ∗ Etera R. Livine, † Matteo Smerlak, ‡ and Simone Speziale § Centre de Physique Th´eorique, CNRS-UMR 6207, Luminy Case 907, 13007 Marseille, France EU Laboratoire de Physique, ENS Lyon, CNRS-UMR 5672, 46 All´ee d’Italie, 69007 Lyon, France EU Perimeter Institute, 31 Caroline St. N., Waterloo, ON N2L 2Y5, Canada (Dated: February 27, 2008)We consider an exact expression for the 6j-symbol for the isosceles tetrahedron, involving SU(2)group integrals, and use it to write the two-point function of 3d gravity on a single tetrahedron as agroup integral. The perturbative expansion of this expression can then be performed with respectto the geometry of the boundary using a simple saddle-point analysis. We derive the completeexpansion in inverse powers of the length scale and evaluate explicitly the quantum correctionsup to second order. Finally, we use the same method to provide the complete expansion of theisosceles 6j-symbol with the explicit phases at all orders and the next-to-leading correction to thePonzano-Regge asymptotics.
Introduction
A wide-spread expectation from a full theory of quantum gravity is the possibility to fix the coefficients appearingin the conventional non-renormalizable perturbative expansion seen as an effective field theory (EFT). To address thisquestion, a necessary tool is to control the perturbative expansion of the full theory. In this paper, we investigatethis issue in the spinfoam formalism, using the 3d toy model with a single dynamical variable introduced in [1] anddeveloped in [2].Pursuing a matching with the EFT, while right at the root of many approaches to quantum gravity, most notablystring theory and the asymptotic safety scenario, has long been obstructed in the spinfoam formalism. This is due tothe difficulty in consistently inserting a background metric to perform the perturbative expansion. The key idea isto relate the n -point functions to the field propagation kernel, via the introduction of a suitable boundary state [3].The boundary state can then be taken to be a coherent state peaked on a classical geometry [5]. We then expectthe boundary geometry to effectively induce a semi-classical background structure in the bulk, which allows to definethe graviton propagator from background-independent correlation functions.The structure of this framework is particularly clear in 3d. Considering for simplicity the Riemannian case, thespinfoam amplitude for a single tetrahedron is the 6j-symbol of the Ponzano-Regge model. Its large spin asymptoticsis dominated by exponentials of the Regge action for 3d general relativity. This is a key result, since the quantizationof the Regge action is known to reproduce the correct free graviton propagator around flat spacetime [6]. The roleof the boundary state is to induce the flat background and to gauge-fix the propagator [7]. Thus the frameworkprovides a clear bridge to Regge calculus as an effective description of spinfoam gravity. However, there is more to it.Indeed, if one works with quantum Regge calculus alone, there are technical problems to go beyond the free theoryapproximation. These are related to the lack of a unique measure for the path integral compatible with the triangleinequalities conditions ensuring that the metric is positive definite. The issue is solved in the spinfoam formalism,where the triangle conditions are automatically imposed on the 6j-symbol by the recoupling theory of SU(2) and themeasure is selected by the topological symmetry of the system. Thus the spinfoam approach does reduce to quantumRegge calculus at leading order but improves it beyond . ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] Similar ideas on the use of coherent states lie also behind the study the semiclassical limit in the canonical loop gravity framework [4]. The situation is more complicated in 4d. Developments of this idea have led to the remarkable result that the Barrett-Crane modelin 4d Riemannian spacetime does reproduce at large scales the scaling behavior of the free graviton propagator (or 2-point function)[8, 9, 10, 11, 12]. This is crucial evidence towards the correctness of the semiclassical limit of LQG. However the same developmentsalso pointed out [5, 12] that the Barrett-Crane model does not reproduce the right tensorial structure of the propagator, thus the modelfails to reproduce General Relativity in the large scale limit. These results have confirmed the validity of the method, and spurred newefforts towards a better understanding of the spinfoam dynamics [13, 14]. This better behaved models should have a semiclassical limit
In this paper we consider the simplest possible setting given by the 3d toy model introduced in [1, 2] and studyanalytically the full perturbative expansion of the 3d graviton. Our results are based on a reformulation of the 6j-symbol and the graviton propagator as group integrals and the saddle point analysis of these integrals. We computeexplicitly the leading order then both next-to-leading and next-to-next analytically and we support these results withnumerical data. Moreover, it was shown in [2] that deviations of the 6j-symbol from the leading order Ponzano-Reggeasymptotics do not contribute to the next-to-leading order of the graviton, but that they enter the next-to-next ordercorrections. Here, the exact representation of the graviton propagator as a group integral naturally incorporates thesedeviations. Finally, an interesting side-product of our calculations is a formula for the next-to-leading order of thefamous Ponzano-Regge asymptotics of the 6j-symbol in the special isosceles configuration.In spite of the simplicity of the model, the framework we develop here has rather generic features useful forcomputing graviton correlation functions in non-perturbative quantum gravity from spinfoam amplitudes, althoughit does not allow us to tackle the more general issue of the existence of a relevant boundary state and of the resultingEFT-like expansion of the correlation functions for a generic spinfoam triangulation. We nevertheless show that thefull perturbative expansion of the two-point function in the spinfoam quantization of 3d gravity is computable. Wehope to apply these same methods and tools to 4d spinfoam models and allow a more thorough study of the fullnon-perturbative spinfoam graviton propagator and correlations in 4d quantum gravity.
I. THE KERNEL AND THE PROPAGATOR AS GROUP INTEGRALSA. The boundary states and the kernel
Let us consider a triangulation consisting of a single tetrahedron. To define transition amplitudes in a backgroundindependent context for a certain region of spacetime, the main idea is to perform a perturbative expansion with respectto the geometry of the boundary. This classical geometry acts as a background for the perturbative expansion. To doso we have to specify the values of the intrinsic and extrinsic curvatures of such a boundary, that is the edge lengthsand the dihedral angles for a single tetrahedron in spinfoam variables. Following the framework set in [1], we restrictattention to a situation in which the lengths of four edges have been measured, so that their values are fixed, say to aunique value j t + . These constitute the time-like boundary and we are then interested in the correlations of lengthfluctuations between the two remaining and opposite edges which are the initial and final spatial slices (see figure1). This setting is referred to as the time-gauge setting. The two opposite edges e and e have respectively lengths j + and j + . In the spinfoam formalism, and in agreement with 3d LQG, lengths are quantized so that j t , j and j are half-integers. j + j + j t + FIG. 1: Physical setting to compute the 2-point function. The two edges whose correlations of length fluctuations will becomputed are in fat lines, and have length j + and j + . These data are encoded in the boundary state of the tetrahedron.In the time-gauge setting, the four bulk edges have imposed lengths j t + interpreted as the proper time of a particlepropagating along one of these edges. Equivalently, the time between two planes containing e and e has been measured tobe T = ( j t + ) / √ The lengths and the dihedral angles are conjugated variables with regards to the boundary geometry, and haveto satisfy the classical equations of motion. Here, it simply means that they must have admissible values to form agenuine flat tetrahedron. Note that the dimension of the SU(2)-representation of spin j , d j ≡ j + 1 is twice the edge given by a modified Regge calculus where the fundamental variables are area and angles, as the one investigated in [15]. length. Setting k e = d je d jt , for e = e , e , the dihedral angles ϑ , ϑ and ϑ t can be expressed in terms of the lengths : ϑ , = 2 arccos (cid:16) k , q − k , (cid:17) and ϑ t = arccos (cid:16) − k k p − k p − k (cid:17) (1)provided k e <
1, a condition ensured by the triangle inequalities. Notice the relation : cos ϑ t = − cos( ϑ ) cos( ϑ ).We then need to assign a quantum state to the boundary, peaked on the classical geometry of the tetrahedron.Since j t is fixed, we only need such a state for e , peaked on the length j + , and for e , peaked on j + . Theprevious works used a Gaussian ansatz for such states. However, it is more convenient to choose states which admita well-defined Fourier transform on SU(2). In this perspective, the dihedral angles of the tetrahedron are interpretedas the class angles of SU(2) elements. As proposed in [2], the Gaussian ansatz can be replaced for the edges e and e by the following Bessel state:Ψ e ( j ) = e − γ e / N h I | j − j e | ( γ e − I j + j e +1 ( γ e i cos( d j α e ) (2)with γ e = d j t (1 − k e ) (3)where N is a normalization coefficient depending on γ e . The functions I n ( z ) are modified Bessel functions of thefirst kind, defined by : I n ( z ) = π R π dφ e z cos φ cos( nφ ), and α e = ϑ e / j e , with γ e as the squared width:Ψ e ( j ) = 1 N r πγ e e − ( j − je )2 γe cos( d j α e ) . (4)The role of the cosine in (2) is to peak the variable dual to j , i.e. the dihedral angle, on the value α e . Then theboundary state admits a well-defined Fourier transform, which is a Gaussian on the group SU(2). We parameterizeSU(2) group elements as g ( φ, ˆ n ) = cos φ + i sin φ ˆ n · ~σ, φ ∈ [0 , π [ , ˆ n ∈ S , where σ i are the Pauli matrices, satisfying σ i = , and φ is the class angle of g . Since the group element g ( φ, ˆ n ) isidentified to g ( φ + π, − ˆ n ), we can restrict φ to live in [0 , π ). The Fourier transform of (2) is then given by: b Ψ e ( φ ) = 12 X η = ± b Ψ ( η ) e ( φ ) (5)with b Ψ ( η ) e = 1 N sin( φ ) sin (cid:0) d j e ( φ + η α e ) (cid:1) e − γ e sin ( φ + η α e ) (6)This state is a class function on SU(2), but b Ψ ( η ) alone is not (due to the φ ↔ − φ symmetry reflecting that a SU(2)group element and its inverse are simply related by conjugation). The semiclassical analysis is crystal-clear : it ispeaked around the angle α e or π − α e , according to the sign of η . The sine shifts the mean length to j e + .These states carry the information about the boundary geometry necessary to induce a perturbative expansionaround it. More precisely, we are interested in the following correlator, W = 1 N X j ′ ,j ′ (cid:26) j ′ j t j t j ′ j t j t (cid:27) O j ( j ′ )Ψ e ( j ′ ) O j ( j ′ )Ψ e ( j ′ ) (7)with O j e ( j ′ ) = 1 d j e (cid:16) d j ′ − d j e (cid:17) (8)where the normalisation factor N is given by the same sum, without the observable insertions O j e ( j ′ e ). W measuresthe correlations between length fluctuations for the edges e and e of the tetrahedron, and it can be interpreted asthe 2-point function for gravity [1], contracted along the directions of e and e .The 6j-symbol, as it enters (7), emerges from the usual spinfoam models for 3d gravity as the amplitude for asingle tetrahedron. In the previous work [2] we studied the perturbative expansion using its well-known (leadingorder) asymptotics in term of the discrete Regge action (for the tetrahedron). Here instead we use the fact that the6j-symbol for the isosceles configuration admits an exact expression as group integrals, (cid:26) j j t j t j j t j t (cid:27) = Z SU(2) dg dg χ j t ( g g ) χ j t ( g g − ) χ j ( g ) χ j ( g ) (9)where χ j ( g ) = sin( d j φ g )sin φ g is the SU(2) character. Then, selecting a specific boundary state as described below, weare able to rewrite also (7) as an integral over SU(2). This allows us to study the perturbative expansion as thesaddle point (or stationary phase) approximation of the integral for large lengths, d j e ≫
1. With respect to [2], thisprocedure has the advantage of including the higher order corrections coming from both the Regge action and thecorrections to the { j } asymptotics. We will come back to this point below.Let us begin by looking at the saddle points of the isosceles 6j-symbol, as the computation of the propagator willhave a similar structure. We first need the angle of the group elements g g and g g − : φ ± = arccos (cid:0) cos φ cos φ ∓ u sin φ sin φ (cid:1) (10)where we used the notation u = ~n · ~n . Then, expanding the rapidly oscillatory phases in exponential form yields: (cid:26) j j t j t j j t j t (cid:27) = 18 π X ǫ ,ǫ ,ǫ +12 ,ǫ − = ± ǫ ǫ ǫ +12 ǫ − Z dφ dφ du f ( φ , φ , u ) e id jt Φ { ǫ } (11)with f ( φ , φ , u ) = sin( φ ) sin( φ )sin( φ +12 ) sin( φ − ) , (12)Φ { ǫ } ( φ , φ , u ) = ( ǫ +12 φ +12 + ǫ − φ − ) + 2 k ǫ φ + 2 k ǫ φ (13)Let us proceed to the search for the stationary points of the phase Φ { ǫ } . The variable u only enters φ ± , and therelated equation, ǫ +12 ∂ u φ +12 + ǫ − ∂ u φ − = 0, is solved by u = ~n · ~n = 0 and ǫ +12 = ǫ − = ǫ . The variational equationswith respect to φ and φ are: d j t (cid:0) ǫ +12 ∂ φ e φ +12 + ǫ − ∂ φ e φ − (cid:1) + d j e ǫ e = 0 e = 1 , , π [ by : ¯ φ = − ǫ ǫ arccos (cid:16) k √ − k (cid:17) + (1 + ǫ ǫ ) π ¯ φ = − ǫ ǫ arccos (cid:16) k √ − k (cid:17) + (1 + ǫ ǫ ) π (15)Notice that this result allows us to give a geometrical interpretation to the class angles entering (11), which is similarto the one for the usual integral formula of the squared 6j-symbol: the isosceles 6j-symbol is peaked on half theinternal, or external, dihedral angles of the classical geometry. Indeed, for example, when ǫ ǫ = −
1, the stationaryangle ¯ φ is ¯ φ = α = ϑ /
2, while for ǫ ǫ = 1, we have ¯ φ = π − α . We perform the complete expansion of the isosceles { j } , using this stationary phase analysis, below in section IV.Now we turn to the graviton propagator. B. The propagator as group integrals
If we insert the expression (9) into (7), the sums over j ′ and j ′ give the SU(2) Fourier transform of the boundarystates. Let us first look at the normalization N : N = Z dg dg χ j t (cid:0) g g − (cid:1) χ j t (cid:0) g g (cid:1) h X j ′ Ψ e ( j ′ ) χ j ′ ( g ) i h X j ′ Ψ e ( j ′ ) χ j ′ ( g ) i (16)= Z dg dg χ j t (cid:0) g g − (cid:1) χ j t (cid:0) g g (cid:1) b Ψ e ( g ) b Ψ e ( g ) (17) For a general configuration, one has to consider the squared 6j-symbol to have an integral expression. This geometric interpretation of the saddle point is not surprising since the 6j-symbol is indeed the unique physical quantum state for atrivial topology and a triangulation made of a single tetrahedron [17]. It satisfies the quantum flatness constraint and can serve as theboundary state in the general boundary framework.
For large spins, we are interested in evaluating N with a saddle point approximation. To that end, let us expand theprevious expression in exponential form : N = 132 π X ǫ ,ǫ ,ǫ +12 ,ǫ − ,η ,η = ± ǫ ǫ ǫ +12 ǫ − Z dφ dφ du f ( φ , φ , u ) e d jt S { ǫ,η } ( φ ,φ ,u ) (18)with S { ǫ,η } ( φ , φ , u ) = i (cid:0) ǫ +12 φ +12 + ǫ − φ − (cid:1) + X e =1 , ik e ǫ e ( φ e + η e α e ) − (cid:0) − k e (cid:1) sin ( φ e + η e α e ) (19)where the label { ǫ, η } refers to the dependence on the sign variables, and f is given by (12). The crucial point is thatthe phase of (18) (the imaginary part of S { ǫ,η } ) is precisely Φ { ǫ } in (13), up to constant α , terms which play norole in the stationary phase approximation. This means in particular that the imaginary part of S { ǫ,η } has the samesaddle points of the isosceles 6j-symbol.The same analysis can be performed for the numerator of W . To take into account the observables O j e , noticethat the SU(2) character is an eigenfunction for the Laplacian on the sphere S with the Casimir as eigenvalue:∆ S χ j ( φ ) = 1sin φ ∂ φ (cid:0) sin φ ∂ φ χ j (cid:1) = − ( d j − χ j ( φ ) . (20)This allows to perform the sums over j ′ and j ′ in (7), introducing the Fourier transforms b Ψ e . Again expanding theresult of these operations into exponential form, one ends up with: W = 132 π N k k ϑ t X ǫ ,ǫ ,ǫ +12 ,ǫ − ,η ,η = ± ǫ ǫ ǫ +12 ǫ − Z dφ dφ du f ( φ , φ , u ) e d jt S { ǫ,η } ( φ ,φ ,u ) × (cid:16) a { ǫ,η } ( φ , φ ) + b { ǫ,η } ( φ , φ ) d j t + c { ǫ,η } ( φ , φ ) d j t (cid:17) (21)The functions a { ǫ,η } , b { ǫ,η } and c { ǫ,η } stand for the observable insertions, and are given by : a { ǫ,η } ( φ , φ ) = Y e =1 , (cid:16) − k e k e sin φ e + η e α e ) − iǫ e sin 2( φ e + η e α e ) (cid:17) (22) b { ǫ,η } ( φ , φ ) = − k cos 2( φ + η α ) (cid:16) − k k sin φ + η α ) − iǫ sin 2( φ + η α ) (cid:17) + (cid:0) e ↔ e (cid:1) (23) c { ǫ,η } ( φ , φ ) = 1 k k cos 2( φ + η α ) cos 2( φ + η α ) (24)We are now ready to study the large spin expansion of (21). A common choice in the literature is to do so usinga power series in 1 /j t (keeping j /j t and j /j t constant). However it is more convenient to take as parameter of theexpansion the dimension d j t (again keeping k , k fixed). This is the natural choice, as we compute correlations withrespect to the background geometry with lengths defined by the half-dimensions d j t / d j / d j /
2. Furthermore,as we show below (see (49) and (51)), these are the values of the lengths emerging in the asymptotics of the 6j-symbol.As written in (21), W corresponds to the mean value of the function a { ǫ,η } + b { ǫ,η } d jt + c { ǫ,η } d jt for the non-lineartheory defined by the action S { ǫ,η } and the integration measure f . The strategy is thus clear: we will computeseparately the normalisation N and the numerator, perturbatively, with an expansion around the saddle points of theaction S { ǫ,η } .As stated above, the imaginary part of S { ǫ,η } has the same saddle points of the isosceles 6j-symbol, namely u = 0and ¯ φ e given in (15) (independently of η and η ). The extremization with respect to the real part of S { ǫ,η } , on theother hand, constrains the η and η signs. Indeed, for a given solution ( ¯ φ , ¯ φ ) from (15), characterized by ǫ , ǫ and ǫ , the signs η and η have to satisfy:sin 2 (cid:0) ¯ φ e + η e α e (cid:1) = 0 , for e = 1 , η = ǫ ǫ and η = ǫ ǫ . This leads to four possibilities, summarized in thefollowing table, η = − η = 1 η = − φ = α , and ¯ φ = α ,ǫ = ǫ = − ǫ ¯ φ = π − α , and ¯ φ = α ,ǫ = − ǫ = − ǫ η = 1 ¯ φ = α , and ¯ φ = π − α , − ǫ = ǫ = − ǫ ¯ φ = π − α , and ¯ φ = π − α ,ǫ = ǫ = ǫ The condition ǫ +12 = ǫ − = ǫ and η = ǫ ǫ and η = ǫ ǫ allows us to perform three sums in (21), theconfigurations for which there is no saddle point being exponentially suppressed : W = 132 π N k k ϑ t X ǫ ,ǫ ,ǫ = ± ǫ ǫ Z dφ dφ du f ( φ , φ , u ) e d jt S { ǫ } ( φ ,φ ,u ) × (cid:16) a { ǫ } ( φ , φ ) + b { ǫ } ( φ , φ ) d j t + c { ǫ } ( φ , φ ) d j t (cid:17) (26)and the same for N without the insertion of k k ϑ t ( a { ǫ } + b { ǫ } /d j t + c { ǫ } /d j t ). Here the label { ǫ } simply indicatesthe dependence of the functions on the signs ǫ , ǫ and ǫ . II. THE COMPLETE PERTURBATIVE EXPANSION
The perturbative expansion of the two-point function W is formulated as an asymptotic power series expansion in1 /d j t , of the type: W = 1 d j t " w + 1 d j t w + 1 d j t w + . . . . (27)Let us remind the reader that the dimension d j t defines the length scale of the tetrahedron L = d j t L P /
2, with thePlanck length L P = ~ G . Such an expansion thus matches the typical expansion of quantum field theory correlationswith quantum corrections ordered in increasing powers of ~ (and of the coupling constant G ) and with L correspondingto the renormalization scale. We can thus call the coefficients w , w , .. the one-loop and two-loop (and so on)corrections.This perturbative expansion is obtained studying the power series expansion in 1 /d j t around each of the four saddlepoints of both the denominator R f exp d j t S { ǫ } and the numerator R ( a { ǫ } + b { ǫ } /d j t + c { ǫ } /d j t ) f exp d j t S { ǫ } . Moreprecisely, we expand the action S { ǫ } around its saddle point: the evaluation of S at the stationary point gives anumerical factor, there is no linear term obviously, the quadratic term defines the Hessian matrix A { ǫ } and finallyall the remaining higher order terms (cubic onwards) are kept together to define the potential Ω { ǫ } . This potentialthus contains all higher order corrections to the quadratic approximation to the action S. As such, it does not enterthe leading order of the two-point function but largely enters its NLO, NNLO and so on, (the loop corrections) as inquantum field theory. Then each term in the power series is evaluated as the Gaussian moment with respect to theHessian matrix A { ǫ } of terms coming from the expansion of f exp d j t Ω { ǫ } in powers of d j t . In general many termsactually contribute to the same overall order in 1 /d j t . This intricate structure comes about precisely as in [2] becausethe expansion of exp d j t Ω { ǫ } gives increasing powers of d j t while the Gaussian moments have increasing powers in1 /d j t .On the other hand, a simplification of our calculations comes from the fact that each saddle point gives the samecontribution. This is a consequence of the symmetry properties of the functions involved under the transformationof φ e into π − φ e . Further, for a given saddle point, the two possible configurations of signs are simply related bycomplex conjugation. The actual sum then ensures the reality of the result. Without loss of generality, we can thusrestrict the computation to the saddle point ( α , α ,
0) with ǫ = 1 = − ǫ = − ǫ . There we have : a ( φ , φ ) = Y e =1 , (cid:16) − k e k e sin φ e − α e ) + 2 i sin 2( φ e − α e ) (cid:17) (28) b ( φ , φ ) = − k cos 2( φ − α ) (cid:16) − k k sin φ − α ) + 2 i sin 2( φ − α ) (cid:17) + (cid:0) e ↔ e (cid:1) (29) c ( φ , φ ) = 1 k k cos 2( φ − α ) cos 2( φ − α ) (30)and the potential Ω is extracted from the derivatives of S greater than three, with S given by (19) with the chosensigns, S ( φ , φ , u ) = i (cid:0) φ +12 + φ − (cid:1) − X e =1 , (cid:0) − k e (cid:1) sin ( φ e − α e ) + linear terms (31)Expanding around the background, the inverse of the Hessian matrix is (see the Appendix A for details): A − = 14 − k cos ϑ t k k e iϑ t cos ϑ t k k e iϑ t − k
00 0 i tan ϑ t − ( k + k ) . (32)Introducing the shorthand notation A − ~β = X all possible pairingsof ( β ,...,β N ) A − β i β i . . . A − β i N − β i N (33)for ~β ∈ { , , } N , the complete perturbative expansion of the propagator can be written as W = k k ϑ t √ − k √ − k d jt P i,j =1 , ∂ ij a A − ij + P P ≥ W P d Pjt P P ∈ N N P d Pjt . (34)In the numerator, the first term gives the leading order contribution in 1 /d j t (see next section). It comes entirelyfrom the a term in (28). In fact, a and b vanish at the saddle point, and so does the gradient of a , so the expansionof a , b and c is dominated by the quadratic term of a .All the higher order corrections have been collected in the summations. The coefficients N P and W P correspondto finite sums: N P = P X n =0 X ~β ∈{ , , } P + n ) P + n ))! n ! ℜ (cid:16) i e − i (2 d jt − ) ϑ t ∂ P + n ) ~β (cid:0) f Ω n (cid:1) A − ~β (cid:17) | φ = α ,φ = α ,u =0 (35)and W P = X n ≥ n ! n X ~β a ∈{ , , } P + n ) P + n ))! ℜ (cid:16) ie − i (2 d jt − ) ϑ t ∂ P + n ) ~β a (cid:0) af Ω n (cid:1) A − ~β a (cid:17) + X ~β b ∈{ , , } P + n − P + n − ℜ (cid:16) ie − i (2 d jt − ) ϑ t ∂ P + n − ~β b (cid:0) bf Ω n (cid:1) A − ~β b (cid:17) + X ~β c ∈{ , , } P + n − P + n − ℜ (cid:16) ie − i (2 d jt − ) ϑ t ∂ P + n − ~β c (cid:0) cf Ω n (cid:1) A − ~β c (cid:17)o | φ = α ,φ = α ,u =0 (36)The three lines of (36) are the three separate contributions of the insertions a , b and c . The sum over n defining W P is finite for each of these contributions : n is bounded by 2 P −
2, 2 P − P − a , b and c respectively. Thederivatives of highest order of Ω involved in W P are respectively the 2 P -th derivatives, the (2 P − P − N P , the 2( P + 1)-th derivatives, all corresponding to n = 1.The intricacy of the formulas was anticipated at the beginning of the section. However the reader should bereassured that they are simple, if tedious, algebraic expressions.The real part ℜ in (35) is consistent with the reality of the initial expression (21), and arises from the summationover the ǫ sign. III. THE LEADING ORDER, ONE-LOOP AND TWO-LOOP CORRECTIONS
We now use (35) and (36) to obtain explicitly the first orders of the expansion. The leading order (LO) and thenext to leading order (NLO) have already been obtained in a quite different way in [2]. We here recover them quickly.The computation of the next to next to leading order (NNLO) is then completely new. It is shown in [2] that theNNLO needs the corrections to the asymptotics of the 6j-symbol, i.e. to the Ponzano-Regge formula. The success ofour method resides in the fact that such corrections are naturally contained in the exact group integral expression (9)of the kernel.The LO is obtained evaluating the normalization at the saddle point, f = f ( α , α , a : W LO = k k ϑ t d jt ℜ (cid:0) ie − i (2 d jt − ) ϑ t ∂ φ ,φ a A − (cid:1) f ℜ (cid:0) ie − i (2 d jt − ) ϑ t (cid:1) (37)= − d j t cos ϑ t sin(2 d j t − ) ϑ t sin(2 d j t − ) ϑ t (38)This reproduces the expected 1 /d j t scaling behavior of the LO. The difference in the coefficient with [1, 2] comesfrom the different boundary state used. In particular notice that while ϑ t ( k , k ) is a constant, the dependence upon d j t of the second fraction produces spurious oscillations. These can be reabsorbed in the boundary state, replacingsin d j e ( φ + η α e ) in b Ψ e with sin (cid:0) d j e ( φ + η α e ) + η d j t ϑ t (cid:1) . The Fourier transform is then:Ψ e ( j ) = e − γ e / N h I | j − j e | ( γ e d j α e + d j t ϑ t ) − I j + j e +1 ( γ e d j α e − d j t ϑ t ) i (39)This does not affect the asymptotic behavior of Ψ e ( j ). With this replacement, we obtain the same result of [2] (cf.equation (37)) for the isosceles case, W LO = − d j t cos ϑ t sin ϑ t sin ϑ t . (40)Even if the LO now matches the previous results presented in [2], the higher orders will differ because of the differentboundary state used.For the sake of a simpler presentation, we will report the NLO and the NNLO for the equilateral tetrahedron, k = k = 1 / ϑ t = arccos − . The general expressions in terms of k and k are indeed quite cumbersome. Thischoice will also facilitate the comparison with numerical simulations of (7).The NLO is then obtained from the coefficients N and W . To keep compact expressions, we adopt the followingsymbolic notation for the contractions of derivatives with Gaussian moments : for functions f and h of φ , φ and u , define: f n h m A − n + m = n ! m ! P i , ··· ,i n =1 , , P j , ··· ,j m =1 , , ∂ ni , ··· ,i n f ∂ mj , ··· ,j m h A − i , ··· ,i n ,j , ··· ,j m ) evaluated at thesaddle point ( α , α ,
0) with ǫ = − ǫ = − ǫ = 1. For example, the LO of (34) can be written d jt a A − . In N ,three powers of Ω appear, Ω , Ω and Ω . Using the boundary state (39), we have: N = ℜ (cid:16) i e i ϑ t h f A − + (cid:0) f S + f S (cid:1) A − + f S S A − i(cid:17) (41)We proceed in the same way for the three contributions to W : W = ℜ (cid:16) i e i ϑ t h(cid:0) a f + a f + f a (cid:1) A − + (cid:0) a f S + f ( a S + a S ) (cid:1) A − + f a S S A − + (cid:0) f b + b f (cid:1) A − + f b S A − + f c i(cid:17) (42)After straighforward algebra we obtain the NLO, of order 1 /d j t : W NLO = 1 d j t − d j t (43)These results for the LO and NLO are well-confirmed by numerical simulations, as one can see from figure 2. Anagreement with 0.58% of error for the LO, and with 1.7% error for the NLO is reached between the coefficients ofthese orders for d j t = 201 (i.e the representation of spin j t = 100).All orders of the expansion can be computed using the above recipe. From this point of view, the NNLO is of noparticular specificity. We need the expansion of the action (or equivalently Ω) until the sixth order. The highest order FIG. 2: Log-log plots comparing numerical simulations with analytical results. Left plot: a numerical simulation of (7)(diamond symbol) compared with the leading order of (43) (star symbol). Middle plot: the next to leading order of (43), instar shape, with the numerics, in diamond shape. Right plot: the next to next to leading order (48). correlator A − ~β is of order 12 for the normalisation, and respectively 14, 10 and 6 for the insertion of a , b and c . N = ℜ (cid:16) i e i ϑ t h f A − + (cid:0) f S + f S + f S + f S (cid:1) A − + 12 (cid:0) f S S + 2 f S S + f S S + 2 f S S (cid:1) A − + 13! (cid:0) f S S S + 3 f S S S (cid:1) A − + f S S S S A − i(cid:17) (44)We also write W = W ( a )3 + W ( b )3 + W ( c )3 , with: W ( a )3 = ℜ (cid:16) i e i ϑ t h(cid:0) f a + f a + f a + f a + f a (cid:1) A − + (cid:0) ( f a + f a + f a + f a ) S +( f a + f a + f a ) S + ( f a + f a ) S + f a S (cid:1) A − + 12 (cid:0) ( f a + f a + f a ) S S +2( f a + f a ) S S + f a ( S S + 2 S S ) (cid:1) A − + 13! (cid:0) ( f a + f a ) S S S +3 f a S S S (cid:1) A − + f a S S S S A − i(cid:17) (45) W ( b )3 = ℜ (cid:16) i e i ϑ t h(cid:0) f b + f b + f b + f b (cid:1) A − + (cid:0) ( f b + f b + f b ) S + ( f b + f b ) S + f b S (cid:1) A − + 12 (cid:0) ( f b + f b ) S S + 2 f b S S (cid:1) A − + f b S S S A − i(cid:17) (46) W ( c )3 = ℜ (cid:16) i e i ϑ t h(cid:0) f c + f c + f c (cid:1) A − + (cid:0) ( f c + f c ) S + f c S (cid:1) A − + f c S S A − i(cid:17) (47)The NNLO is thus computed to be: W NNLO = 1 d j t − d j t + 520507157464 d j t (48)This result is again supported by numerical simulations, see figure 2. An agreement with 11.3% of error is obtainedfor the coefficient of 1 /d j t at d j t = 201. The error can be reduced pushing the simulations to higher values of d j t . IV. PERTURBATIVE EXPANSION OF THE ISOSCELES 6J-SYMBOL
The procedure described above can be applied directly to the isosceles 6j-symbol (9), obtaining the known Ponzano-Regge formula and its corrections. This is interesting for a number of reasons. As discussed in [2], the corrections tothe Ponzano-Regge formula are a key difference between the spinfoam perturbative expansion studied here, and theone that would arise from quantum Regge calculus. The 6j-symbol is also the physical boundary state of 3d gravityfor a trivial topology and a one-tetrahedron triangulation. In 4d, it appears as a building block for the spin-foamsamplitudes, such as the 15j-symbol. Thus, with regards to many aspects of spin-foams in 3d and 4d, in particular forthe quantum corrections to the semiclassical limits, it would be good to have a better understanding of this object0beyond the Ponzano-Regge asymptotics. This is what we do in this section, performing a perturbative expansion ofthe exact expression (9) for the isosceles 6j-symbol. Indeed, it is a simpler application of the procedure developedabove for the propagator.As written in (11), the 6j-symbol is the partition function for the theory defined by the action Φ { ǫ } and theintegration measure f . Notice first of all that the Regge action of 3d gravity S R = P d j e ϑ e naturally appears as theevaluation of Φ { ǫ } at the saddle points: ( e id jt Φ { ǫ } ( ¯ φ , ¯ φ , = ǫ ǫ e − iǫ ǫ ǫ S R S R = d j t (cid:0) ϑ t + 2 k α + 2 k α (cid:1) (49)We then proceed exactly as for the propagator, knowing that for each configuration of signs, the { j } is peaked onthe classical geometry of the tetrahedron. The perturbative expansion with respect to this flat geometry is thus givenby the Gaussian moments of the Hessian matrix H { ǫ } of Φ { ǫ } . Let us stress that, in contrast with the previous studiesof the asymptotics of the 6j-symbol, we are here scaling the lengths of the tetrahedron (or equivalently d j t ), keepingthe length ratios k and k fixed, instead of scaling j t .As for the propagator, the four saddle points give the same contribution, and the two sign configurations of agiven saddle point are related by complex conjugation. This can be done in a quite explicit way. Introduce ω tobe the truncated Taylor expansion of Φ { ǫ } , starting at order three onwards, around the saddle point ( α , α ,
0) with ǫ = − ǫ = − ǫ = 1. Let H − be the corresponding inverse of H { ǫ } : H − = 12 − k cot ϑ t − √ − k √ − k sin ϑ t − √ − k √ − k sin ϑ t − k cot ϑ t
00 0 tan ϑ t − ( k + k ) (50)We also introduce the volume of the tetrahedron, which enters the Gaussian integrals of H : V t = d j t k k q − ( k + k ) . (51)The expansion of this isosceles 6j-symbol is then (see appendix.B for more details): ( j j t j t j j t j t ) = 1 p − k p − k √ πV t X p ≥ ( − p (cid:16) C p d pj t cos (cid:0) S R + π (cid:1) + C p +1 d p +1 j t sin (cid:0) S R + π (cid:1)(cid:17) (52)where the coefficients C P , for P = 2 p, p + 1, are given by finite sums: C P = P X n =0 ( − n (2( P + n ))! n ! X ~β ∈{ , , } P + n ) ∂ P + n ) ~β (cid:16) f ω n (cid:17) | ( α ,α , H − ~β (53)Thus, all even orders are in phase with the leading order asymptotics, given by the original Ponzano-Regge for-mula in cos( S R + π/ C , with f ( α , α ,
0) = p − k p − k : ( j j t j t j j t j t ) ∼ LO √ πV t cos (cid:16) S R + π (cid:17) . (54)On the other hand, all odd orders are out of phase (or in quadrature of phase) with this leading order. If we werescaling the spin j t instead of the length d j t /
2, the result would not have had such a simple structure with sines andcosines being mixed up at all orders (but leading).This asymptotic series formula for the isosceles tetrahedron shows that only the Regge action is relevant and noother frequency appears in the 6j-symbol. We believe this feature to generalize to the generic 6j-symbol since itsasymptotics can also be extracted using saddle point techniques [16].The coefficient of a given order is simply given by the contractions of the derivatives of f ω n with the Gaussianmoments. For a given order P , the highest derivatives of ω involved correspond to n = 1 in (53) and equals 2( P + 1).For instance, the NLO is obtained by setting P = 1. With the notations of the previous section, we have: C = f H − − ( f ω + f ω ) H − + f ω ω H − (55)1
200 400 600-0.008-0.006-0.004-0.002 2000 3000 4000 5000-0.0225-0.0175-0.015-0.0125-0.01-0.0075-0.005 6000 8000 10000120001400016000-1.75-1.5-1.25-0.75-0.5
FIG. 3: Differences between the 6j-symbol and the analytical result (56) for three pairs ( k , k ) = ( k, k ): from left to right, k = 1 / k = 3 / k = 3 /
42. The X axis stands for d j t = Nd j , for d j respectively fixed to 1, 7, and 21, while N goes from200 to 800. and introducing the reduced volume v = V t /d j t : ( j j t j t j j t j t ) ∼ NLO √ πV t cos (cid:16) S R + π (cid:17) − cos ϑ t d j t √ πV t P ( k , k )48(12 v ) sin (cid:16) S R + π (cid:17) , (56)where P ( k , k ) is a symmetric polynomial in k and k : P ( k , k ) = 3(1 − k ) (1 − k ) + 3(1 − k ) (1 − k ) − k k + 25 k k − k k + k k ) + 10( k k + k k ) . (57)This polynomial is not simply related to the volume and we haven’t succeeded in providing it with a geometricinterpretation. It would nevertheless be very interesting to understand its geometrical origin in order to interpretphysically the higher order corrections to the graviton propagator.For extremal values of k , this polynomial simplifies. We get P (0 , k ) = 3(1 − k ) (1 − k ) for k = 0. At the otherend at k = 1, we obtain P (1 , k ) = − k (1 − k ) with obvious roots 0 and 1. Let us point out that k , actuallynever physically reaches these extreme values 0 and 1, but its bounds depend on the representation j t (due to theSU(2) recoupling theory): 12 d j t ≤ k e ≤ − d j t . When k reaches these extreme values, the coefficients of P are polynomials in 1 /d j t . The result (56) is confirmed by numerical simulations, see figure 3. These plots represent numerical simulations ofthe { j } minus the analytical formula (56), for three pairs ( k , k ). We have used in these simulations the particularcase k = k = k , for which P ( k, k ) = (1 − k ) (cid:16) − k + 55 k − k (cid:17) (58)whose only root in [0 ,
1] is k = q √
310 + 1450)) + (10(81 √ − + 55] ≈ . d / j t to see how the coefficient of theNLO is approached, and suppressed the oscillations by dividing by those of the NNLO, cos( S R + π ). The numericssupport both the coefficient and the phase.Notice that in the equilateral situation, k = k = 1 / For instance, when the edge e is at minimal length, j = 0 or k = d jt , the coefficients of P read: P ( 12 d j t , k ) = 3 ` − d j t + 516 d j t − d j t ´ + ` −
12 + 232 d j t − d j t + 532 d j t ´ k + ` − d j t + 2516 d j t ´ k + ` − d j t ´ k . /j t expansion: { j t } NLO = 2 / q πd j t cos (cid:0) S R + π (cid:1) − · / q πd j t sin (cid:0) S R + π (cid:1) (59)= cos (cid:0) S R + π (cid:1) / p πj t − / p πj t h (cid:0) S R + π (cid:1) + 31 · /
576 sin (cid:0) S R + π (cid:1)i (60)This point of view shows that it is much more natural to study the asymptotics of the 6j-symbol in term of the inverselength 1 /d j instead of the inverse spin label 1 /j . For instance, the leading order coefficient is given in term of thevolume V t of the tetrahedron with edge lengths given by the d j ’s and not the j ’s.Finally, we point out that the asymptotics given above in term of the cosine and sine of the Regge action holds formid-range values of k , k and it breaks down for k , k close to their extremal values 0 and 1. Indeed when k = 0 theasymptotics are given in term of Airy functions while when k = 1 they are given by the (non-oscillatory) exponentialof the Regge action. The interested reader can find details and references in [17]. Conclusions
We have shown it is possible to compute analytically the two-point function – the graviton propagator – at all ordersin the Planck length for the 3d toy model (the Ponzano-Regge model for a single isoceles tetrahedron) introduced in[1]. This builds on the previous work [2] where the leading order and first quantum corrections were computed usingthe asymptotics of the 6j-symbol in term of the Regge action. Here, we introduced a representation of the relevant6j-symbol and of the full graviton propagator as group integrals over SU(2). Then one obtains the expansion of thetwo-point function as a power series in the inverse spin label (or equivalently in the Planck length) by expandingthese group integrals around their saddle points. We computed explicitly the first and second order corrections to theleading order behavior and matched them successfully against numerical simulations.A side-product of these calculations is the corrections to the Ponzano-Regge asymptotic formula for the 6j-symbolfor an isosceles tetrahedron (when four representations are taken equal). We obtain a series alternating cosines andsines of the Regge action for the tetrahedron (shifted by π/ P ( k , k ) in front of this first order correction.To conclude, we have shown how to carry out the calculations of the spinfoam graviton propagator at all orders atleast in this simple setting. We hope to apply the present methods and tools to more refined 3d triangulations [18]and to compute spinfoam correlations for 4d quantum gravity along the lines of [9, 11, 12, 13, 19]. Acknowledgements
The plots and numerical data presented here were computed using Mathematica 5.0. MS is grateful for thehospitality of the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by theGovernment of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research& Innovation.
APPENDIX A: DETAILS FOR THE PROPAGATOR EXPANSION
The key object containing the quadratic fluctuations and their corrections is the generating function Z ( J ), whichis the Gaussian integral of the Hessian matrix A with a source J , evaluated at each saddle point: Z { ǫ } ( J ) = Z dX e − djt XA { ǫ } X + JX , (A1)with X = ( φ , φ , u ). The Hessian matrix is given, for all configurations of signs, by: A { ǫ } = − k sin ϑ t e iǫ ǫ ǫ ( π − ϑ t ) k k sin ϑ t cos ϑ t e − iǫ π k k sin ϑ t cos ϑ t e − iǫ π − k sin ϑ t e iǫ ǫ ǫ ( π − ϑ t )
00 0 − iǫ ǫ ǫ (cid:2) − ( k + k ) (cid:3) cos ϑ t sin ϑ t . (A2)3Due to the initial symmetry between the groups elements g and g , clearly expressed in (17), A { ǫ } is invariant underthe reversing of ǫ and ǫ . A { ǫ } has also the property of transforming into its complex conjugated matrix whenreverting all signs, an operation which does not change the saddle point ( ¯ φ , ¯ φ ,
0) considered, and equivalently underthe flipping of ǫ . A straightforward calculation yields: Z { ǫ } ( J ) = Z { ǫ } e djt JA − { ǫ } J (A3)with Z { ǫ } = (cid:16) πd j t (cid:17) / − iǫ ǫ ǫ (1 − k )(1 − k ) p | cos ϑ t | e i ǫ ǫ ǫ ϑ t , (A4)and A − { ǫ } = 14 − k ǫ ǫ ϑ t k k e iǫ ǫ ǫ ϑ t ǫ ǫ ϑ t k k e iǫ ǫ ǫ ϑ t − k
00 0 − ( k + k ) tan ϑ t e iǫ ǫ ǫ π . (A5) A − { ǫ } and Z { ǫ } benefit from the previously mentioned symmetries of A { ǫ } . The symmetry flipping ǫ and ǫ meansthat the saddle points ( α , α ,
0) and ( π − α , π − α ,
0) have the same Hessian matrices, for a fixed ǫ . Moreover,flipping ǫ while going to the saddle points ( α , π − α ,
0) and ( π − α , α ,
0) does not change A { ǫ } and Z { ǫ } , up toa change of sign for the non-diagonal coefficients of A { ǫ } .Let us focus on the normalization N . The Gaussian moments are generated by successive derivations of Z ( J ) withrespect to the source, and they are contracted with the derivatives of f exp d j t Ω { ǫ } , which we expand into powers of d j t . We thus have: N = 132 π X ǫ ,ǫ ,ǫ ǫ ǫ Z { ǫ } e d jt S { ǫ } ( ¯ φ , ¯ φ , X N ∈ N ~β ∈{ , , } N X n ≥ N )! n ! d N − nj t ∂ N~β (cid:0) f Ω n { ǫ } (cid:1) | ¯ φ , ¯ φ ,u =0 A − { ǫ } ,~β (A6)where the correlators A − { ǫ } ,~β are defined according to Wick’s theorem: A − { ǫ } ,~β = X all possible pairingsof ( β ,...,β N ) A − { ǫ } ,β i β i . . . A − { ǫ } ,β i N − β i N (A7)As Ω is a Taylor expansion into powers of ( φ − ¯ φ ), ( φ − ¯ φ ) and u , whose minimal order is 3, the power n of Ωin (A6) is bounded from above by N : 3 n ≤ N , and the sum over n is thus finite for each N . The power of 1 /d j t receives two contributions: one, positive, from the Gaussian moments, and the other, negative from the expansion ofexp d j t Ω { ǫ } . We can identify the coefficients of a given order by the simple change of variables P = N − n . Introducingthe explicit expressions of Z { ǫ } and S { ǫ } ( ¯ φ , ¯ φ , N = − − k )(1 − k ) p π | cos ϑ t | d / j t X P ∈ N u P d Pj t (A8)with u P = P X n =0 X ~β ∈{ , , } P + n ) X ǫ ,ǫ ,ǫ iǫ ǫ ǫ e − iǫ ǫ ǫ (2 d jt − ) ϑ t P + n ))! n ! ∂ P + n ) ~β (cid:0) f Ω n { ǫ } (cid:1) | ¯ φ , ¯ φ ,u =0 A − { ǫ } ,~β (A9)Let us further simplify the coefficients u P by performing the sums over ǫ , ǫ and ǫ . First, notice that the sign ofthe imaginary part of Ω { ǫ } is ǫ ǫ ǫ . Since f is real, and considering the symmetry properties of A − { ǫ } given in (A5),it is clear that when the signs ǫ , ǫ and ǫ are all flipped, the derivatives are evaluated at the same saddle point and u P is transformed into its complex conjugate. Thus, let us work with a fixed value of ǫ ǫ ǫ , say 1, and consider thebasic properties of the functions f and S { ǫ } minus the linear parts in φ and φ (its derivatives greater than threeare those of Ω). More precisely, we are interested in how these functions and their derivatives, evaluated at a givensaddle point, transform when the saddle point is changed. Let us see for instance the differences when going betweenthe saddle points ( ¯ φ = α , ¯ φ = α ) and ( ¯ φ = π − α , ¯ φ = α ).Having impose the value of ǫ ǫ ǫ , this change of saddle point is determined by the flips of ǫ and ǫ . We have: f ( π − φ , φ , u ) = f ( φ , φ , u ) and ( φ +12 + φ − )( π − φ , φ , u ) = 2 π − ( φ +12 + φ − )( φ , φ , u ), while the real part of S { ǫ } isnon-zero only when derivated an even number of times. Thus, f Ω n ( φ , φ , u ) equals f Ω n ( π − φ , φ , u ) when we flipin the same time ǫ in front of ( φ +12 + φ − ) in S { ǫ } . This means that each derivation with respect to φ flips the signbetween the two saddle points considered. There is now three possibilities: (i)such a derivation is contracted with4another derivation w.r.t. φ through A − = − k ) , then the sign is changed twice, i.e. there is no change of sign.(ii)It is contracted with a derivation w.r.t. u via A − u which is zero, so that there is in fact no contribution. (iii)It iscontracted with a derivation w.r.t. φ via A − whose sign changes under the flip of ǫ . Thus these two saddle pointsgive the same contribution. The proof can be repeated between the four saddle points. Finally: N = − − k )(1 − k ) p π | cos ϑ t | d / j t X P ∈ N N P d Pj t (A10)with N P given by (35).The same analysis can be performed for the numerator of the propagator. One has simply to take into accountthe fact that the insertion k k ϑ t ( a { ǫ } + b { ǫ } /d j t + c { ǫ } /d j t ) involves three different powers of d j t . To perform thesums over the signs ǫ , ǫ and ǫ , first notice, like for the denominator, that flipping of all them three transforms thecoefficients into its complex conjugate. Then, restricting attention to a fixed value of the product ǫ ǫ ǫ , it is easy tocheck that the derivative of a , b and c w.r.t. φ evaluated at φ = π − α is equal to ( − p times that evaluated at φ = α , while flipping ǫ and ǫ , with p being the number of derivatives w.r.t. φ . The same is true for φ . Thus,we can reproduce the previous argument showing that the four saddle points give the same contribution. This leadsus to: W = − k k − k )(1 − k ) cos ϑ t p π | cos ϑ t | N d / j t n d j t X i,j =1 , ∂ ij a A − ij + X P ≥ W P d Pj t o (A11)with W P given by (36). APPENDIX B: DETAILS FOR THE EXPANSION OF THE 6J-SYMBOL
Let us compute the generating function: Z (6 j ) { ǫ } ( J ) = Z dX e − djt X e H { ǫ } X + JX (B1)with e H { ǫ } = 2 iǫ sin ϑ t ǫ ǫ (1 − k ) cos ϑ t p − k p − k p − k p − k ǫ ǫ (1 − k ) cos ϑ t
00 0 − ǫ ǫ (cid:2) − ( k + k ) (cid:3) cos ϑ t . (B2)Taking care of the fact that e H { ǫ } has purely imaginary coefficients, one has: Z (6 j ) { ǫ } ( J ) = Z (6 j ) { ǫ } e djt J e H − { ǫ } J (B3)with Z (6 j ) { ǫ } = π p − k p − k √ πV t e − iǫ ǫ ǫ π , (B4)and e H − { ǫ } = iǫ ǫ ǫ − k cot ϑ t − √ − k √ − k sin ϑ t − √ − k √ − k sin ϑ t ǫ ǫ − k cot ϑ t
00 0 ǫ ǫ − ( k + k ) tan ϑ t , (B5)where the volume V t is given by (51). Using (49), we obtain an expression similar to (A8): ( j j t j t j j t j t ) = 18 p − k p − k √ πV t X P ≥ e C P d Pj t (B6)with the series coefficients in term of the Hessian: e C P = P X n =0 P + n ))! n ! X ǫ ,ǫ ,ǫ ( iǫ ) n e − iǫ ǫ ǫ ( S R + π ) X ~β ∈{ , , } P + n ) ∂ P + n ) ~β (cid:0) f ω n { ǫ } (cid:1) | ¯ φ , ¯ φ ,u =0 e H − { ǫ } ,~β (B7)5We are now in position to repeat the arguments of the previous section. The symmetries of the functions f , iǫ ω ,combined with those of e H − { ǫ } imply that the four saddle points contribute the same. Moreover, the two configurationsof signs corresponding to a given saddle point, which are related by flipping ǫ , ǫ and ǫ , are related by complexconjugation. The coefficient e C P is thus completely determined by the saddle point ( α , α ,
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