Maïté Dupuis
email: maite.dupuis@ens-lyon.fr

Etera R Livine
email: etera.livine@ens-lyon.fr

The 6j-symbol: Recursion, Correlations and Asymptotics

Keywords:

We study the asymptotic expansion of the {6j}-symbol using the Schulten-Gordon recursion relations. We focus on the particular case of the isosceles tetrahedron and we provide explicit formulas for up to the third order corrections beyond the leading order. Moreover, in the framework of spinfoam models for 3d quantum gravity, we show how these recursion relations can be used to derive Ward-Takahashi-like identities between the expectation values of graviton-like spinfoam correlations.

I.

different correlation functions of different orders. These equations play a key role in the study of the renormalization of quantum field theories, so we expect these new recursion relations to be equally relevant to the study of the renormalization/coarse-graining of spinfoam models.

II. THE RECURSION RELATION FOR THE ISOSCELES TETRAHEDRON

A. Exact and Approximate Recursion

We will focus on the isosceles tetrahedron, which is relevant for the computation of geometrical correlations in the simplest non-trivial toy model in 3d quantum gravity [START_REF] Speziale | Towards the graviton from spinfoams: the 3d toy model[END_REF]. Such a tetrahedron has four of its edges of equal length with the two remaining opposite edges of arbitrary length. The corresponding isosceles {6j}-symbol is:

{a, b} J ≡ a J J b J J ,
where J ∈ N/2 and a, b are integers smaller than 2J (to satisfy the triangular inequality). The associated tetrahedron has edge lengths l j = d j /2 for j = a, b, J, where d j = 2j + 1 is the dimension of the SU(2) representation of spin j.

The volume V of the tetrahedron is given by the simple formula:

V J (a, b) = 1 12 l a l b 4l 2 J -(l 2 a + l 2 b ), (1)
while the (exterior) dihedral angles θ j can also be written in term of the edge lengths (see e.g. [START_REF] Bonzom | Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model[END_REF] for more details):

cos θ a = - 4l 2 J -l 2 a -2l 2 b 4l 2 J -l 2 a , cos θ b = - 4l 2 J -l 2 b -2l 2 a 4l 2 J -l 2 b , cos θ J = -l a l b 4l 2 J -l 2 a 4l 2 J -l 2 b . (2)
The general recursion relation for the {6j}-symbol given by Schulten and Gordon in [START_REF] Schulten | Exact Recursive Evaluation of 3J and 6J Coefficients for Quantum Mechanical Coupling of Angular Momenta[END_REF] simplifies in this specific isosceles case:

(l a + 1 2 ) 4l 2 J -(l a + 1 2 ) 2 {a + 1, b} J -2l a (4l 2 J -l 2 a ) cos(θ a ) + 1 4 {a, b} J + (l a - 1 2 ) 4l 2 J -(l a - 1 2
) 2 {a -1, b} J = 0

(3) In the asymptotic regime, we know (analytically and numerically) the behavior of the {6j}-symbol at the leading order:

{a, b} J ∼ {a, b} LO J ≡ 1 √ 12πV cos l a θ a + l b θ b + 4l J θ J + π 4 , (4)
which is actually valid under the assumption that the tetrahedron with edge lengths l a , l b , l J exists (else the generic asymptotics can be expressed in term of Airy functions). The oscillatory phase is given by the Regge action S R = l a θ a + l b θ b + 4l J θ J . Using the obvious trigonometric identity cos((n + 1)φ) + cos((n -1)φ) = 2 cos φ cos nφ, we can write an exact recursion relation for the leading order of the {6j}-symbol:

V J (a + 1, b) {a + 1, b} LO J -2 cos θ a V J (a, b) {a, b} LO J + V J (a -1, b) {a -1, b} LO J = 0. (5)
A similar recursion relation holds for b-shifts and also J-shifts.

The most natural idea is to compare this recursion relation for the leading order to the previous equation on the exact {6j}-symbol to see how to use them to extract the next-to-leading correction to the asymptotic behavior. We can first find the link between the leading order of equation [START_REF] Freidel | A New Spin Foam Model for 4d Gravity[END_REF] and the leading order of equation [START_REF] Rovelli | Graviton propagator from background-independent quantum gravity[END_REF]. Both equations can be written in the same form at the leading order:

{a + 1, b} J -2 cos θ a {a, b} J + {a -1, b} J ≈ 0, (6)
which turns into a simple second order differential equation in the large spin limit. Then the next-to-leading order of the exact recursion relation given in equation ( 3) is easily computed,

V J (a, b) 1 + 1 2la 1 - 2l 2 a 4l 2 J -l 2 a {a + 1, b} J -2 cos θ a V J (a, b){a, b} J + V J (a, b) 1 -1 2la (1 - 2l 2 a 4l 2 J -l 2 a ) {a -1, b} J ≈ 0, (7)
and will have to be compared to a recursion relation for the next-to-leading order approximation of the {6j}-symbol.

Here, we have kept the extra-factor V J (a, b) to underline the similarity of this relation with the leading order recursion relation [START_REF] Rovelli | Graviton propagator from background-independent quantum gravity[END_REF].

B. Pushing to the Next-to-Leading Order

We are interested in the asymptotic expansion of the {6j}-symbol. It was shown in previous works [START_REF] Bonzom | Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model[END_REF][START_REF] Dupuis | Pushing Further the Asymptotics of the 6j-symbol[END_REF] that l j seems to be the right parameter to consider when studying the semi-classical behavior of the {6j}-symbol. So from now we write:

a J J b J J ≡ {l a , l b } lJ .
Notice that shifting a by ±1 is equivalent to shifting the edge length l a = a + 1/2 by ±1. We rescale now l j by λl j and we replace the exact {6j}-symbol by a series in 1/λ alternating cosines and sines of the Regge action (shifted by π/4) in the previous equation [START_REF] Freidel | A New Spin Foam Model for 4d Gravity[END_REF]. The fact that there is no mixing up of cosines and sines at any order was show in [START_REF] Bonzom | Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model[END_REF]. More precisely, we write the {6j}-symbol asymptotic expansion in the form:

{λl a , λl b } λlJ = 1 λ 3/2 D(l a , l b , l J ) cos(λS R + π/4) (8)
+ 4 n=1 F (n) (l a , l b , l J ) λ n sin(λS R + π/4) + G (n) (l a , l b , l J ) λ cos(λS R + π/4)) + O(λ -5 ) ,
where the pre-factor denominator D(l a , l b , l J ) is given by the square-root of the tetrahedron volume as in equation ( 4).

To study the asymptotics, it is convenient to factorize the whole equation ( 3) by λ 3/2 . We then write {l a ± 1/λ, l b } lJ for {λl a ± 1, λl b } λlJ . We also factorize the coefficients of the recursion relation. We start by defining

C(l j ) = l a (4l 2 J -l 2 a ) = 16(A(la,lJ ,lJ )) 2 la where A(a, b, c) = 1 4 (a + b + c)(a + b -c)(a -b + c)(-a + b + c
) is the area of the triangle of edge lengths given by a, b and c. The coefficient which appears in front of

{l a ± 1/λ, l b } lJ becomes C(l a ± 1/(2λ), l b , l J ) = (l a ± 1/(2λ))(4l 2 J -(l a ± 1/(2λ)))
, where we underline that the shift is ±1/(2λ) and not simply ±1/λ. We expand C(l a ± 1/(2λ), l b , l J ) in term of derivatives:

C(l a ± 1/(2λ), l b , l J ) = n 1 n! 1 (2λ) n ∂ n C ∂l n a with              C = l a (4l 2 J -l 2 a ) ∂C ∂la = 4l 2 J -3l 2 a ∂ 2 C ∂l 2 a = -6l 2 a ∂ 3 C ∂l 3 a = -6 ∂ n C ∂l n a = 0 for n ≥ 4 (9)
Then to express {l a ± 1/λ, l b } lJ we need to expand D(l a ± 1/λ), F (i) (l a ± 1/λ), and

G (i) (l a ± 1/λ): (i ∈ {1 • • • 4})        D(l a ± 1/λ) = D ± 1 λ ∂D ∂la + 1 2λ 2 ∂ 2 D ∂l 2 a ± 1 3!λ 3 ∂ 3 D ∂l 3 a + 1 4!λ 4 ∂ 4 D ∂l 4 a F (i) (l a ± 1/λ) = 4-i k=0 (-1) k 1 k!λ k ∂ k F (i) ∂l k a G (i) (l a ± 1/λ) = 4-i k=0 (-1) k 1 k!λ k ∂ k G (i) ∂l k a (10)
F (1) (l j ) was computed in a previous paper [START_REF] Bonzom | Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model[END_REF][START_REF] Dupuis | Pushing Further the Asymptotics of the 6j-symbol[END_REF]. It was also suggested that the asymptotic expansion of the {6j}-symbol in terms of the length scale λ is given by a series alternating in cosines and sines at each order, so we expect that G (i) (l j ) = 0 for ∀i ≥ 1. Finally, we also need to expand the Regge action λS R (l a ± 1 λ ), remembering that

θ j = θ j (l a ) : λS R (l a ± 1 λ ) = λS R + 4 k=0 (-1) k+1 (k + 1)!λ k ∂ k θ a ∂l k a (11) with                              ∂θa ∂la = -2lal b (4l 2 J -l 2 a ) √ 4l 2 J -l 2 a -l 2 b ∂ 2 θa ∂l 2 a = - 2l b (4l 2 J l 2 a -2l 4 a -l 2 a l 2 b +16l 4 J -4l 2 b l 2 J ) (4l 2 J -l 2 a ) 2 [4l 2 J -l 2 a -l 2 b ] 3/2 ∂ 3 θa ∂l 3 a = - 2lal b (24l 4 b l 2 J +40l 2 J l 2 a l 2 b -12l 2 J l 4 a +5l 4 a l 2 b -192l 4 J l 2 a -240l 4 J l 2 b +2l 2 a l 4 b +6l 6 a +576l 6 J ) (4l 2 J -l 2 a ) 3 [4l 2 J -l 2 a -l 2 b ] 5/2 ∂ 4 θa ∂l 4 a = 1 (4l 2 J -la 2 ) 4 (4l 2 J -l 2 a -l 2 b ) 7/2 (6(8l 10 a + 8l 8 a l 2 b + 152l 2 J l 6 a l 2 b -720l 4 J l 6 a + 7l 6 a l 4 b + 3520l 6 J l 4 a -1472l 4 J l 4 a l 2 b + 2l 4 a l 6 b +140l 4 a l 4 b l 2 J -560l 2 a l 4 b l 4 J -3840l 8 J l 2 a + 2432l 6 J l 2 a l 2 b + 48l 2 a l 2 J l 6 b + 2048l 8 J l 2 b -448l 6 J l 4 b -3072l 10 J + 32l 6 b l 4 J )l b )
We can now write an asymptotic recursion equation from equations (3), ( 9), ( 8), [START_REF] Roberts | Classical 6j-symbols and the tetrahedron[END_REF] and [START_REF] Bonzom | Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model[END_REF] in terms of λ neglecting terms of order O(λ -4 ) and smaller, assuming that λ is large. This leads to a couple of equations at each order, one for the cos-oscillations and one for the term in sin:

• The first equation is given by the terms of order λ 0 and it is trivially satisfied (0 = 0) since we have already written the leading order of the {6j}-symbol proportional to cos(S R + π 4 ) (the Ponzano-Regge asymptotic formulae). • The second equation is given by the terms of order λ -1 :

1 2C ∂C ∂l a - 1 D ∂D ∂l a sin(θ a ) + 1 2 ∂θ a ∂l a cos(θ a ) = 0 (12)
which can be rewritten as a differential equation for D:

∂ ln D ∂l a = 1 2 ∂θ a ∂l a cos θ a sin θ a + ∂ ln C ∂l a . (13)
This allows to determine D: ln D = 1 2 ln(C sin(θ a )) + K, which simplifies into D = K l a l b 4l 2 J -l 2 a -l 2 b where K is a constant factor. Thus this second equation shows that D is as expected proportional to the square-root of the volume V of the isosceles tetrahedron. To determine the normalization constant K (as well as G (1) ), the orthonormality property of {6j}-coefficients can be employed: a 4l a √ l b l b {a, b} J {a, b } J = δ bb and we get the K =

√

12π. The details are given in the next section.

• The third equation is given by the terms of order λ -2 and which are proportional to cos(S R + π 4 )

∂F (1) ∂la = la

4C sin θa + 1 2C ∂C ∂la -1 D ∂D ∂la 1 2 ∂θa ∂la + 1 6 ∂ 2 θa ∂l 2 a + cos θa sin θa 1 2D ∂ 2 D ∂l 2 a -1 8C ∂ 2 C ∂l 2 a + 1 D ∂D ∂la 1 2C ∂C ∂la -1 D ∂D ∂la + ( 1 2 ∂θa ∂la ) 2 (14)
where we used the fact that

1 2C ∂C ∂la -1 D ∂D ∂la sin(θ a ) + 1 2 ∂θa ∂la cos(θ a ) = 0 (eqn. ( 13
)
) to remove all the terms proportional to F (1) itself. The first term of the right-hand side of the equation ( 14) comes from the variation of the coefficient in front of {a, b} J in the recursion equation ( 3). The terms with a derivative of C with respect to l a come from the coefficients in front of {a ± 1, b} J and {a, b} J . The variation of C with respect to l a is given by the variation of the areas of the triangles of the tetrahedron. From eqn.( 13), we relate it to the variations of D (the volume) and to the variations of the dihedral angle θ a : 1

C ∂C ∂la = 2 D ∂D ∂la -cos θa sin θa ∂θa ∂la .
The terms with a derivative of D with respect to l a come from the variation of the leading order of the asymptotic of the {6j}symbol and the terms with a derivative of the dihedral angle θ a come from the variations of the Regge action S R . We can now compute the derivative of F (1) with respect of l a (equation ( 14)) in terms of l a , l b and l J the edge lengths of the tetrahedron:

∂F (1) ∂la = - 1 48(l 2 a (-4l 2 J +l 2 a ) 2 (4l 2 J -l 2 a -l 2 b ) (5/2) l b ) (-32l 6 b l 2 J l 2 a + 10l 6 b l 4 a + 96l 6 b l 4 J -960l 6 J l 4 b + 15l 6 a l 4 b + 400l 4 J l 4 b l 2 a -100l 2 J l 4 b l 4 a -168l 2 J l 6 a l 2 b -1664l 6 J l 2 b l 2 a + 20l 8 a l 2 b + 576l 4 J l 2 b l 4 a + 3072l 8 J l 2 b -3072l 10 J + 48l 4 J l 6 a + 2304l 8 J l 2 a -576l 6 J l 4 a ) (15
) and then easily integrate this equation over l a :

F (1) (l j ) = - 768l 6 J (l 2 J -l 2 a -l 2 b )+736l 4 J l 2 a l 2 b +240l 4 J (l 4 a +l 4 b )-176l 2 J l 2 a l 2 b (l 2 a +l 2 b )-24l 2 J (l 6 a +l 6 b )+10l 2 a l 2 b (l 4 a +l 4 b )+25l 4 a l 4 b 24(4l 2 J -l 2 b )(2l 2 J -l 2 a )(4l 2 J -l 2 b -l 2 a ) 3/2 lal b + Z(l b , l J ) (16)
The integration constant Z(l b , l J ) can be determined using the symmetry properties of the {6j}-symbol: symmetry of the isosceles {6j}-symbol with respect to l a and l b , coupling of l a , l b and l J by this isosceles {6j}-symbol and homogeneity of F (1) ([F (1) ] = l -1 j ) imply that Z(l b , l J ) = 0. Then this gives us the same result as in the previous paper [START_REF] Dupuis | Pushing Further the Asymptotics of the 6j-symbol[END_REF]. Moreover, using the definitions of the tetrahedron volume (1) and of the dihedral angles (2), we can express F (1) in terms of some geometrical characteristics of the tetrahedron:

F (1) = - cos θ J 3(12V ) 8 -(12V ) 4 l 4 a l 4 b 3(l 2 a -l b ) 2 + 2l 2 a l 2 b -l 12 a l 12 b + 6l 12 a l 12 b 48(12V ) 3 l 8 a l 8 b (17)
• The fourth equation is given by the terms of order λ -2 and which are proportional to sin(S R + π 4 ). It is the same equation as the previous one for G (1) but the right-hand side is now equal to zero (homogenous equation). That is we simply get that ∂G (1) ∂l a = 0 (18) so G (1) = Z(l b , l J ) is just a constant of integration. Once again the symmetry properties of the {6j}-symbol implies that G (1) = 0.

• The next equation is given by the terms of order λ -3 and which are proportional to sin(S R + π 4 ). We get an equation for the first derivative of F (2) with respect to l a ∂F (2) (lj ) ∂la

= cos θa 2 sin θa ∂ 2 F (1) ∂l 2 a - 1 sin 2 θa ∂θa ∂la + F (1) ∂F (1) ∂la + cos θa sin θa -1 4! ∂ 3 θa ∂l 3 a + 1 D ∂D ∂la 1 2C ∂C ∂la + 1 2D ∂ 2 D ∂l 2 a -1 D ∂D ∂la 2 -1 8C ∂ 2 C ∂l 2 a + 1 3! 1 2 ∂θa ∂la 2 1 2 ∂θa ∂la + 1 D ∂D ∂la -1 2C ∂C ∂la 1 3! ∂ 2 θa ∂l 2 a + 1 2C ∂C ∂la 1 2D ∂ 2 D ∂l 2 a + 1 8C ∂ 2 C ∂l 2 a 1 D ∂D ∂la + 1 2C ∂C ∂la -1 D ∂D ∂la 1 2 1 2 ∂θa ∂la 2 + 1 3! ∂ 2 θa ∂l 2 a 1 2 ∂θa ∂la + 1 D ∂D ∂la 3 -1 D ∂D ∂la 1 D ∂ 2 D ∂l 2 a + 1 3!D ∂ 3 D ∂l 3 a -1 8•3!C ∂ 3 C ∂l 3 a -1 2C ∂C ∂la 1 D ∂D ∂la 2 (19)
We recall that D is proportional to the square root of the tetrahedron volume, C can be expressed in terms of the volume V and the sinus of the dihedral angle θ a (see equation ( 13)). To integrate this equation, we first express it explicitly 1 in terms of l a , l b and l J , and then deduce F (2) :

F (2) (l j ) = -1 4608((4l 2 J -l 2 a ) 2 (4l 2 J -l 2 b ) 2 (4l 2 J -l 2 a -l 2 b ) 3 l 2 a l 2 b ) (-2359296l 2 a l 10 J l 4 b -224512l 6 a l 6 J l 4 b + 100l 12 a l 4 b + 576l 4 J l 12 b + 112896l 8 J l 8 b +2727936l 4 a l 12 J + 5308416l 16 J + 212l 10 b l 6 a -5898240l 2 a l 14 J -11520l 10 a l 6 J + 941056l 4 a l 8 J l 4 b + 31584l 8 a l 4 J l 4 b -2416l 4 a l 10 b l 2 J -79872l 8 a l 6 J l 2 b -480l 12 b l 2 J l 2 a -7040l 8 a l 6 b l 2 J -2416l 10 a l 2 J l 4 b + 100l 4 a l 12 b + 212l 10 a l 6 b +2727936l 12 J l 4 b -700416l 10 J l 6 b -5898240l 14 J l 2 b -11520l 6 J l 10 b -700416l 6 a l 10 J + 609l 8 a l 8 b + 112896l 8 a l 8 J +576l 12 a l 4 J -2359296l 4 a l 10 J l 2 b + 528384l 6 a l 8 J l 2 b + 5849088l 2 a l 12 J l 2 b -79872l 2 a l 8 b l 6 J + 31584l 4 a l 8 b l 4 J -7040l 6 a l 8 b l 2 J -224512l 4 a l 6 b l 6 J + 58816l 6 a l 6 b l 4 J + 8640l 10 a l 4 J l 2 b + 8640l 10 b l 4 J l 2 a -480l 12 a l 2 J l 2 b + 528384l 2 a l 8 J l 6 b ) (20
) which is the only result with the required symmetries 2 . The geometrical meaning of this function does not seem obvious. Nevertheless, we can give a more compact expression for the denominator of F (2) :

(4l 2 J -l 2 a ) 2 (4l 2 J -l 2 b ) 2 (4l 2 J -l 2 a -l 2 b ) 3 l 2 a l 2 b = (12V ) 6 cos 4 θ J . ( 21
)
1 ∂F (2) ∂la (l j ) = -

1 2304((4l 2 J -l 2 b )l 3 a (4l 2 J -l 2 a ) 3 (4l 2 J -l 2 a +l 2 b ) 4 l 2 b ) (-1604l 8 a l 8 b l 2 J + 1250816l 4 a l 8 J l 6 b -207104l 6 a l 6 b l 6 J + 31904l 10 a l 4 J l 4 b -169344l 4 a l 8 b l 6 J - 3920l 6 a l 10 b l 2 J + 24992l 6 a l 8 b l 4 J -46848l 10 a l 6 J l 2 b -7129088l 4 a l 10 J l 4 b + 1770496l 6 a l 8 J l 4 b + 16832l 4 a l 10 b l 4 J + 34368l 8 a l 4 J l 6 b -6816l 12 a l 4 J l 2 b -278912l 8 a l 6 J l 4 b + 14524416l 2 a l 12 J l 4 b + 486144l 2 a l 8 J l 8 b -560l 12 b l 4 a l 2 J -43776l 2 a l 10 b l 6 J -3317760la 2 lJ 10 l 6 b + 22241280l 4 a l 12 J l 2 b + 794l 12 a l 6 b -6955008l 6 a l 10 J l 2 b + 2801664l 12 J l 6 b + 672l 14 a l 2 b l 2 J -26542080l 4 a l 14 J -451584l 10 J l 8 b + 46080l 8 J l 10 b -2304l 12 b l 6 J -21233664l 18 J + 37158912l 2 a l 16 J + 1072128l 8 a l 8 J l 2 b - 1528la 12 l 4 b l 2 J -10911744l 14 J l 4 b -35979264l 2 a l 14 J l 2 b + 1728l 12 b l 4 J l 2 a + 9953280l 6 a l 12 J + 228096l 10 a l 8 J -10368l 12 a l 6 J -2073600l 8 a l 1 J 0 + 27l 10 a l 8 b + 23592960l 16 J l 2 b + 400l 8 a l 10 b -88l 14 a l 4 b -8144l 10 a l 6 b l 2 J + 100l 12 b l 6 a )
2 If the result is not symmetric after integration, a non-null integration constant has to be added and its determination can be done using the symmetry properties of the {6j}-symbol. Indeed, we have ∂F (2) ∂la = H(la, l b , l J ) so by integration over la, F (2) (l j ) = h(la, l b , l J )+Z(l b , l J ). Moreover by symmetry, we must have ∂F (2) ∂l b = H(la = l b , l b = la, l J ) and then integrating over l b , we obtain a second expression for F (2) : F (2) (l j ) = h(la = l b , l b = la, l J ) + Z(la, l J ) which implies that the constant of integration satisfies Z(l b , l J ) -Z(la, l J ) = h(la = l b , l b = la, l J ) -h(la, l b , l J ). This equation allows to determine Z and to get (20).

• The next equation comes from the terms of order λ -3 which are proportional to cos(S R + π 4 ):

∂G (2) ∂l a (l j ) = 0 (22)
which implies once again that G (2) = Z(l b , l J ) is a constant of integration. Then the symmetry properties of the {6j}-symbol implies G (2) (l j ) = 0.

We can now give the asymptotic expansion of an isosceles {6j}-symbol until the next to next to leading order (NNLO):

{l a , l b } NNLO lJ = 1 12πV lJ (l a , l a ) cos(S R + π 4 ) + F (1) (l j ) sin(S R + π 4 ) + F (2) (l j ) cos(S R + π 4 ) ( 23
)
where the expression for F (1) and F (2) are given by equations ( 17) and (20). This result seems to confirm that the expansion of the {6j}-symbol is a series alternating in cosines and sines of the Regge action (shift by π 4 ). In the case of an equilateral tetrahedron, all the edges have the same length, that is l a = l b = l J = l and V = √ 2 12 l 3 . Then equation (23) reduces to:

{6j} NNLO equi = 1 πl 3 √ 2 cos(S R + π 4 ) - 31 72 2 1/4 2 5/2 √ πl 5 sin(S R + π 4 ) - 45673 20736 1 2 1/4 2 4 √ πl 7 cos(S R + π 4 ) ( 24
)
where the Regge action is given by S R = 6lθ and θ = θ a = θ b = θ J = arccos(-1/3). This result is confirmed by numerical simulations. The plot figure 1 represents numerical simulations of the equilateral {6j}-symbol minus its approximation (24). Moreover, to enhance the comparison, we have multiplied by l 7/2 to see how the coefficient of the NNLO is approached and we have divided by sin(S R + π 4 ) (oscillations of the next to next to next to leading order) to suppress the oscillations. This gives an error that decreases as expected as l -1 .

FIG. 1: Difference between the equilateral {6j}-symbol and the analytical result (24). The x-axis stands for l = d/2 and d goes from 100 to 5000. The error decreases as expected as l -1 confirming our asymptotic formula.

• The next two equations come from the terms of order λ -4 . The equation for G (3) is the same as the one for G (1) and G (2) , that is: ∂G (3) ∂la = 0. Using the same arguments of symmetry we deduce that G (3) = 0. This confirms our expectation of a series alternating in cosines and sines in the asymptotic of the {6j}-symbol:

{λl a , λl b } λlJ = 1 λ 3/2 D(l a , l b , l J ) cos(λS R + π 4 ) + ∞ k=1 F (k) (l a , l b , l J ) λ k cos(λS R + π 4 + (k) π 2 ) , (25)
where (k) = -1 when k is odd and (k) = 0 when k is even. We already have an expression for F (1) and F (2) . The equation for F (3) is the second equation of order λ -4 and gives its first derivative with respect to l a in terms of l a , l b and l J . It is straightforward (though lengthy) to integrate it over l a and get the expression 3 of F (3) in terms of l a , l b and l J . In the equilateral case (l a = l b = l J = l), the formula reduces to:

F (3) = 28833535 17915904 1 2 9/2 2 1/4 l 3 (26)
Therefore, we have the expression of the asymptotic expansion of the equilateral {6j}-symbol up to the nextto-next-to-next to leading order (NNNLO):

{6j} NNNLO equi = 1 2 1/4 √ πl 3 cos(S R + π 4 ) - 31 72 2 5/2 l sin(S R + π 4 ) - 45673 20736 2 4 l 2 cos(S R + π 4
) + 28833535 17915904 2 9/2 l 3 sin(S R + π 4

) .

We check this result numerically by computing it using Mathematica for two values of spins j = 50 and j = 100. More precisely, we computed the renormalized error

({6j}equi-{6j} NNNLO equi )l 9/2 cos(SR+ π 4 )
and we got the expected 1/λ-behavior. However for l > 100, Mathematica is not accurate enough and the numerical errors are too important to get exploitable results. In the general isosceles case, the expression of F (3) is quite complicated and its geometrical interpretation remains to be understood. Nevertheless, we can again give as before a more compact formula for the denominator of F (3) :

denominator F (3) = 3317760(4l 2 J -l 2 a -l 2 b ) 9/2 l 3 a l 3 b (4l 2 j -l 2 a ) 3 (4l 2 J -l 2 b ) 3 = 30(48) 3 (12V J (a, b)) 9 cos 6 θ J ( 28
)
From this equation, the equation giving the denominator of F (2) and remembering that the denominator of F (1) can be written in a similar form: denominator F (1) = 48 (12V ) 3 cos 2 θJ , we can conjecture that:

denominator

F (k) ∝ (12V J (a, b)) 3k (cos θ J ) 2k (29)
where F (k) are the terms appearing in the asymptotic expansion of the {6j}-symbol (25). And consequently, the numerator of F (k) is a polynomial in l j of degree 8k.

So, using the recursion relation for the isosceles {6j}-symbol as well as its symmetry properties, we have computed explicitly the asymptotic expansion of the isosceles {6j}-symbol to the fourth order up to an overall factor K (this integration constant K comes from the integration of the first equation ( 13)). The well-known value K = √ 12π which already appears in the Ponzano-Regge formula can be obtained easily using the unitary property of the {6j}-symbol, as we show in the next section. The equilateral case has been checked against numerical calculations. This method using the recursion relation is fairly easy to implement. It requires integrating a rational fraction at each level and involves neither a Riemann sum nor a saddle point analysis. Moreover, since the coefficient C of the recursion relation ( 3) is a polynomial of degree 3; ∂ n C ∂l n a = 0 for n ≥ 4. Therefore, we expect to get a stable relation for the first derivative of F (k) and G (k) with respect to l a for k ≥ 3. On the one hand, this allows one to prove that G (k) always vanishes; and on the other hand, it should provide a systematic method to extract F (k) for arbitrary order k.

We conclude this section with a general remark on the asymptotic expansion of the {6j}-symbol. In the context of 3d quantum gravity, it is often argued that the leading order of the {6j}-symbol is a cos(S) instead of a complex phase exp(+iS), thus reflecting that the path integral is invariant under a change of (local) orientation (see e.g. [START_REF] Freidel | Discrete Space-Time Volume for 3-Dimensional BF Theory and Quantum Gravity[END_REF]). This obviously neglects the +π/4 shifts, which can be considered as a quantum effect (like an ordering ambiguity). However, in the light of the present expansion, it is clear that we have terms of the type sin(S) beyond the leading order and such terms are not invariant under the change S → -S. This means that the role of this symmetry in the spinfoam path integral should be more subtle than originally thought.

3 F (3) (l j ) = - 1 3317760(l 3 a (la-2l J )(la+2l J )(4l 2 J -l 2 b ) 3 (4l 2 J -l 2 a -l 2 b ) ( 9/2)l 3 b (4l 2 J -l 2 a ) 2 ) (-6965741568l 6 a l 10 J l 8 b + 1069728l 6 a l 16 b l 2 J + 28270080l 16 a l 6 J l 2 b - 743040l 4 J l 2 a l 18 b +1750503260160l 20 J l 2 a l 2 b +379404800l 12 a l 6 J l 6 b +788705280l 10 a l 10 J l 4 b -33547200l 6 a l 14 b l 4 J -98578608l 12 a l 8 b l 4 J -81032970240l 8 a l 14 J l 2 b + 379404800l 6 a l 12 b l 6 J -389214720l 8 J l 2 a l 14 b -323813376000l 6 a l 18 J -156036l 8 a l 16 b -22176l 2 J l 4 a l 18 b -3824640l 16 a l 4 J l 4 b -28408l 18 a l 6 b -31104000l 16 a l 8 J - 1262270545920l 18 J l 2 a l 4 b -1262270545920l 18 J l 4 a l 2 b + 461814497280l 16 J l 2 a l 6 b + 1069728l 16 a l 2 J l 6 b + 74144841728l 6 a l 12 J l 6 b + 824520278016l 16 J l 4 a l 4 b + 461814497280l 6 a l 16 J l 2 b + 788705280l 10 J l 4 a l 10 b -267541217280l 6 a l 14 J l 4 b + 115174656l 6 J l 4 a l 14 b -999532800l 8 J l 4 a l 12 b + 68611276800l 8 a l 16 J - 3824640l 4 J l 4 a l 16 b -1242078720l 10 a l 8 J l 6 b +28270080l 6 J l 2 a l 16 b +38483472384l 12 J l 4 a l 8 b +2157096960l 12 a l 10 J l 2 b -1242078720l 6 a l 10 b l 8 J +509607936000l 24 J - 33547200l 14 a l 6 b l 4 J + 1222041600l 12 J l 2 a l 10 b + 11119152l 12 a l 10 b l 2 J -999532800l 12 a l 8 J l 4 b + 783522201600l 20 J l 4 a + 2157096960l 10 J l 2 a l 12 b -28408l 6 a l 18 b - 389214720l 14 a l 8 J l 2 b -783601920l 8 a l 8 J l 8 b -98578608l 8 a l 4 J l 12 b -6965741568l 8 a l 10 J l 6 b -323813376000l 18 J l 6 b +1222041600l 10 a l 12 J l 2 b +115174656l 14 a l 6 J l 4 b - 743040l 18 a l 4 J l 2 b -22176l 18 a l 2 J l 4 b -81032970240l 14 J l 2 a l 8 b + 7201368l 14 a l 8 b l 2 J -114011616l 10 a l 10 b l 4 J -985242009600l 22 J l 2 b + 568565760l 10 a l 8 b l 6 J - 1401753600l 12 a l 12 J -267541217280l 14 J l 4 a l 6 b +568565760l 8 a l 6 J l 10 b +11119152l 10 a l 12 b l 2 J +783522201600l 2 J 0l 4 b -342523l 12 a l 12 b +38483472384l 8 a l 12 J l 4 b - 3981312000l 10 a l 14 J -156036l 16 a l 8 b -511602l 10 a l 14 b +344217600l 14 b l 10 J -1401753600l 12 b l 12 J +1036800l 18 b l 6 J +344217600l 14 a l 10 J +68611276800l 16 J l 8 b - 3981312000l 14 J l 10 b -511602l 14 a l 10 b -31104000l 16 b l 8 J -985242009600l 22 J l 2 a + 1036800l 18 a l 6 J + 7201368l 8 a l 14 b l 2 J )

C. Consequences of the unitary property of the {6j}-symbols

The orthogonality property of the {6j}-symbols states that:

la 4l a l b l b {l a , l b } lJ {l a , l b } lJ = δ l b l b (30)
This relation corresponds to the unitarity of the evolution in the Ponzano-Regge 3d quantum gravity. We want to use this property to determine the constant of the leading order of the {6j}-symbol. cos(S R + π 4 ); but K is still undetermined. For large spin and for l b ≈ l b , we can approximate the unitary property at the leading order in (l b -l b ) by:

∞ 0 dl a 4l a l b K 2 V J (a, b) cos(S R (l a , l b ) + π 4 ) cos(S R (l a , l b ) + π 4 ) ≈ δ(l b -l b ). ( 31
)
The product of the cosines can be simplified at leading order:

cos(S R (l a , l b ) + π 4 ) cos(S R (l a , l b ) + π 4 ) = 1 2 cos(S R (l a , l b ) + S R (l a , l b ) + π 2 ) + cos(S R (l a , l b ) -S R (l a , l b )) ∼ 1 2 cos(2S R (l a , l b ) + π 2 ) + cos((l b -l b )θ b ) ,
where the dihedral angle θ b = arccos -

4l 2 J -l 2 b -2l 2 a 4l 2 J -l 2 b
is considered as a function of the length l a . We do a saddle point approximation. The first term oscillates and its integral is exponentially suppressed. And we are left with the second term, which should satisfy the following equation:

∞ -∞ dl a l a l b K 2 V J (a, b) cos((l b -l b )θ b ) ≈ δ(l b -l b ) ( 3 2 )
We recall that:

1 2π ∞ -∞ dl a cos(l a (l b -l b )) = δ(l b -l b )
therefore we can conclude that

l a l b K 2 V = 1 2π ∂θ b ∂l a . ( 33
)
θ b and l b are conjugate variables and K comes from the Jacobian of the change of variables between l a and θ b .

Computing the derivative of the dihedral angle gives:

∂θ b ∂l a = -2 4l 2 J -l 2 a -l 2 b = -l a l b 6V J (a, b) ⇒ K = 1 √ 12π . (34)
Moreover, pushing the approximation of the unitary property to the next to leading order in (l b -l b ) and using the next to leading order of the {6j}-symbol shows that G (1) = 0. This was already shown in the previous part using the recursion relation and the symmetry properties of the {6j}-symbol and comes as a confirmation.

III. "WARD-TAKAHASHI IDENTITIES" FOR THE SPINFOAM GRAVITON PROPAGATOR

We are interested in the two-point function in 3d quantum gravity for the simplest triangulation given by a single tetrahedron. This provides the first order of the "spinfoam graviton propagator" in 3d quantum gravity.

Considering the isosceles tetrahedron, we focus on the correlations between the two representations a and b:

O(a) O(b) ψJ = 1 Z a,b ψ J (a)ψ J (b)O(a) O(b){a, b} J , Z ≡ a,b ψ J (a)ψ J (b){a, b} J , (35)
where ψ J (j) is the boundary state, which depends also on the bulk length scale J, and O, O are the observables whose correlation we are studying. Now, inserting a recursion relation with shifts on a, b or J in the sum over the representation labels a,b leads to equations relating the expectation values of different observables. We distinguish two cases: when the state ψ J does not change or when the length scale J also varies.

A. Relating Observables

Inserting the recursion relation on a-shifts in the definition of the correlation function, we obtain the following exact identity:

ψJ (a-1) ψJ (a) O(a -1) O(b)(l a -1 2 )(4l 2 J -(l a -1 2 ) 2 ) ψ -O(a) O(b)2l a (2 cos θ a (4l 2 J -l 2 a ) + 1 4 ) ψ + ψJ (a+1) ψJ (a) O(a + 1) O(b)(l a + 1 2 )(4l 2 J -(l a + 1 2 ) 2 ) ψ = 0. (36)
We call this a Ward identity for our spinfoam correlation. If the observable diverges at a = 0, more precisely if it contains terms in 1/a or in 1/(a + 1), then we need to take into account extra boundary terms in this equation corresponding to contributions at a = 0. But all observables usually considered are regular in this sense.

Then one can choose different sets of observables O and O and one gets different identities on the correlation functions of the spinfoam model. For example, taking O(a) = l a , we get:

ψJ (a-1) ψJ (a) O(b)(l a -1)(l a -1/2)(4l 2 J -(l a -1/2) 2 ) ψ -O(b)(2 cos θ a l 2 a (4l 2 J -l 2 a ) + l 2 a /2) ψ + ψJ (a+1) ψJ (a) O(b)(l a + 1)(l a + 1/2)(4l 2 J -(l a + 1/2) 2 ) ψ = 0.
We recall that the area of the triangle of edge lengths given by l a , l J , l J is equal to

A(l a , l J ) = 1 4 l a 4l 2 J -l 2 a ; then (l a ± 1)(l a ± 1/2)(4l 2 J -(l a ± 1/2) 2 ) = 16[A 2 (l a ± 1/2, l J ) ± A 2 (la±1/2,lJ ) 2(la±1/2)
], therefore we can rewrite the previous equation as an equation between correlation functions of the observable O(b) and different observables proportional to the square of the triangle area A(l a , l J ):

ψJ (a-1) ψJ (a) [A 2 (l a -1/2, l J ) -A 2 (la-1/2,lJ ) 2(la-1/2) ] O(b) ψ -(2 cos θ a A 2 (l a , l J ) + l 2 a /2) O(b) ψ + ψJ (a+1) ψJ (a) [A 2 (l a + 1/2, l J ) + A 2 (la+1/2,lJ ) 2(la+1/2) ] O(b) ψ = 0.
The standard choice of boundary is a phased Gaussian [START_REF] Rovelli | Graviton propagator from background-independent quantum gravity[END_REF][START_REF] Speziale | Towards the graviton from spinfoams: the 3d toy model[END_REF][START_REF] Livine | Physical boundary state for the quantum tetrahedron[END_REF]:

ψ J (j) ∼ e i2lj ϑ e -2α (l j -l J ) 2 l J , ( 37
)
where ϑ is a fixed angle defining a posteriori the external curvature of the boundary and α is an arbitrary real positive number (which can be fixed by the requirement of a physical state [START_REF] Livine | Physical boundary state for the quantum tetrahedron[END_REF]). In this case, we can compute explicitly the ratio ψ(a ± 1)/ψ(a) entering the Ward identity:

ψ J (a ± 1) ψ J (a) = e ±i2ϑ e ∓4α la -l J l J e -2α l J
Of course, this ratios does not depend on b; therefore if the observable O(b) = 1, then the dependence on b only appears in one correlation function through the cosine of the dihedral angle θ a . As another example, we consider

O(a) = l -1 a and O(b) = 4l 2 J -l 2 b (2lJ ) 4 , then: ψJ (a-1) ψJ (a) la-1/2 la-1 4l 2 J -(la-1/2) 2 4l 2 J 4l 2 J -l 2 b 4l 2 J ψ -2 cos θ a 4l 2 J -l 2 a 4l 2 J 4l 2 J -l 2 b 4l 2 J + 1 16l 2 J 4l 2 J -l 2 b 4l 2 J ψ + ψJ (a+1) ψJ (la) la+1/2 la+1 (4l 2 J -(la+1/2) 2 4l 2 J 4l 2 J -l 2 b 4l 2 J ψ = 0
which can be approximated by:

e -i2ϑ e 4α la-l J l J ∆((l a -1/2) 2 )∆(l 2 b ) ψ -2e 2α

B. Rescaling the Tetrahedron

We can now vary also the length scale l J . First let's notice that in the same way we wrote an exact recursion relation for the leading order of the isosceles {6j}-symbol shifting the representation a (equation ( 5)), we can write a similar exact recursion relation for the leading order of the {6j}-symbol shifting the label J; that is (40)

We hope that such equations will turn out useful to study the asymptotic properties of the correlations function as the length scale J grows large, but we leave this for future investigation.

Conclusion

We have used the recursion relation satisfied by the {6j}-symbol to study the structure of its asymptotical expansion for large spins. The exact recursion relation allowed us to compute explicit the asymptotical approximation of the isosceles {6j}-symbol up to fourth order. This confirms previous results [START_REF] Bonzom | Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model[END_REF][START_REF] Dupuis | Pushing Further the Asymptotics of the 6j-symbol[END_REF] and introduces techniques allowing further systematic analytical calculations of the corrections to the behavior of the{6j}-symbol at large spins. A clear and simple geometrical interpretation of the polynomials appearing in this expansion is still missing. The differential equations that we provide for these coefficients are nevertheless a first step in this direction.

This work is useful in particular for the study of large scale correlations in the spinfoam model for 3d quantum gravity. In this context, the recursion relation allowed us to write equations satisfied by the spinfoam correlations similar to the Ward identities of standard quantum field theory. We hope that such recursion techniques can be further applied to the study of 4d spinfoam amplitudes and the resulting spinfoam graviton propagator [START_REF] Bonzom | Recursion relations for spinfoam amplitudes[END_REF].

  From the recursion relation we have shown that {l a , l b } LO lJ = K √ VJ (a,b)

l J cos θ a ∆(l 2 a

 2

V3l J - 2 (3l J - 2 (

 22 J+1 (a, b){a, b} LO J+1 -2 cos(4θ J ) V J (a, b){a, b} LO J + V J-1 (a, b){a, b} LO J-1 = 0 (38)Inserting this recursion relation on J-shifts in the definition correlation function, we obtain the following identity:V J+1 (a, b) ψ J (a)ψ J (b) ψ J+1 (a)ψ J+1 (b) O(a) O(b) ψ + V J-1 (a, b) ψJ (a)ψJ (b) ψJ-1(a)ψJ-1(b) O(a) O(b) ψ -2 cos(4θ J ) V J (a, b)O(a) O(b) ψ = 0 (39)The correlation functions appearing in this equation are in fact approximations. We are allowed to use the leading order of the {6j}-symbol because the boundary state used picks the function on large j 0 . And for the same reason, we can expand V J±1 (a, b) and the ratios ψJ (a)ψJ (b) ψJ±1(a)ψJ±1(b) :V J (a, b) 1 -la +l b ) 2l J (2l J -la-l b ) ] O(a) O(b) ψ -2 cos(4θ J ) V J (a, b)O(a) O(b) ψ + V J (a, b) la +l b ) 2l J (2l J -la -l b ) ] O(a) O(b) ψ ≈ 0.

Ackowledgements

The numerical simulations and plots were done using Mathematica 5.0. MD and ER are partially supported by the ANR "Programme Blanc" grant LQG-06.


Pietro Menotti
email: menotti@df.unipi.it

On the semiclassical treatment of Hawking radiation

Keywords: 04.60.-m, 11.15.Kc, 45.20.Jj Hawking, semiclassical

In the context of the semiclassical treatment of Hawking radiation we prove the universality of the reduced canonical momentum for the system of a massive shell self gravitating in a spherical gravitational field within the Painlevé family of gauges.

We show that one can construct modes which are regular on the horizon both by considering as Hamiltonian the exterior boundary term and by using as Hamiltonian the interior boundary term. The late time expansion is given in both approaches and their time Fourier expansion computed to reproduce the self reaction correction to the Hawking spectrum.

Introduction

In paper [1] Kraus and Wilczek introduced a semiclassical treatment of Hawking radiation by considering the mechanics of a thin self-gravitating shell of matter in a spherical gravitational field. The interest of such an approach lies in the fact that contrary to the usual external field treatment, energy conservation is taken into account which allows to compute the self energy correction to the Hawking formula.

In [1,2] the connection with Hawking radiation was obtained by interpreting the exponential of the classical action as the modes of the system. In this way in the first approximation the well known Hawking result was re-obtained and in the full semiclassical approximation, corrections to the Hawking formula due to the self-energy were computed.

In this analysis the key role is played by the imaginary part of the canonical momentum which appears in the reduced Hamiltonian.

Subsequently [3] the results obtained in [1,2] were given a new interpretation as describing a tunneling phenomenon. Alternative formulae and criticisms were proposed in this context [4,5,6,7,8,9,10,11,12]. In the present paper however we shall go back to the computation of the Bogoliubov coefficients as originally done in [1,2,13].

In [14,15] the problem of the dynamics of one thin shell of matter in the framework of [1,16] was examined critically and a precise definition of the canonical momentum was given through a limit process extending the treatment also to a massive shell of matter.

The procedure for deriving the reduced Hamiltonian in [1], which is obtained in implicit form, is usually considered as very complicated. In [17] it was shown that by introducing a proper generating function it is possible to drastically simplify such a derivation and that the procedure works also for massive shells. The method allows also to extend the treatment to any finite number of massive or massless shells which can cross in their time development. Moreover, it was shown that the expression of the canonical momentum obtained in [1,14] holds in a more general setting and that no limit procedure is necessary for obtaining it.

In this paper we shall examine two aspects of the problem: The first is the universal character of the canonical momentum which appears in the reduced Hamiltonian within the class of the Painlevé gauges. This is done in Sec. 2. Then we shall point out how the shell dynamics can be obtained by considering as Hamiltonian either the exterior mass or an interior mass which appear in the boundary terms. This is done in Sec. 3.

In Sec. 4 we revisit the extraction of the Bogoliubov coefficients from the semiclassical modes. A general treatment along this line was given in the paper [13], but here we shall go back to the explicit late-time development of the modes. We show that in constructing the semiclassical modes which are regular at the horizon one can use either the exterior mass as Hamiltonian as was done originally in [1,13] or an "interior" mass. The two Hamiltonians are related to two different times, the exterior time, which is the usual Killing time t at space infinity, and the interior time which we shall denote by t . Both approaches are consistent and they produce the same result for the absolute value of the ratio of the β to the α Bogoliubov coefficients. In Sec. 5 we give a discussion of the main conclusions. In Appendix A we give some details of the calculations.

The reduced action

In this section we recall some features of the reduced action in the Painlevé family of gauges. The spherically symmetric metric is written as

ds 2 = -N 2 dt 2 + L(dr + N r dt) 2 + R 2 dΩ 2 (1)
where the functions N, N r , L, R can be consistently assumed to be continuous functions of r and t [14]. In [17] the function

F = RL R L 2 -1 + 2M R + RR log   R L - R L 2 -1 + 2M R   (2)
was introduced which has the remarkable property of generating the conjugate momenta through the variational derivatives

π L = ∂F ∂L - ∂ ∂r ∂F ∂L = ∂F ∂L , π R = ∂F ∂R - ∂ ∂r ∂F ∂R .
M denotes the quantity

M = π 2 L 2R + R 2 - R(R ) 2
2L 2 which as a consequence of the constraints is constant in r above and below the shell position r [16,1,17]. The value of M below and above r will be denoted by M and H respectively. The Painlevé family of gauges is identified by the choice L = 1 in the metric (1) where the generating function F assumes the form

F = RW (R, R , M) + RR (L(R, R , M) -B(R, M)) (3)
where

W (R, R , M) = R 2 -1 + 2M R , L(R, R , M) = log(R -W (R, R , M)) (4)
and

B(R, M) = 2M R + log(1 - 2M R ). (5)
In going from eq.( 2) to eq.( 3) we exploited the freedom of adding a total derivate to F with the result that the function (3) has the useful property of vanishing whenever R = 1.

One is still left with the freedom to impose a gauge condition on R. In [17] several choices were examined which can be characterized by a deformation function, of bounded support around the shell position r R(r) = r + h(rr).

We shall call a gauge of the outer type if h(x) = 0 for x ≥ 0; if h(x) = 0 for x ≤ 0 the gauge is called of inner type. There are however other choices in which h is not vanishing in a neighborhood of 0 both for positive and negative argument. The mathematical nature of the deformation h(x) is specified before eq.( 7).

It is useful to start from the general form of the action on a bounded region of space-time as given in [18] to which the shell action

S shell = t f t i dt p ṙ
is added and using the generating function F , one can derive in a straightforward way [17] the reduced action, boundary terms included

t f t i p c ṙ -Ṁ (t) r(t) r 0 ∂F ∂M dr + (-N r π L + NRR )| rm r 0 dt (6)
where the outer gauge g(x) = 0 for x ≥ 0 has been adopted in the Painlevé family of gauges. The boundary terms are just the one given in the paper [18] computed for the spherically symmetric problem at hand. They are equivalent to

-HN(r m ) + MN(r 0 )
where as a consequence of the constraints M and H are constant in r except at the position of the shell r. M is the interior mass while H denotes the exterior mass. Furthermore as a consequence of the gravitational equations combined with the constraints, M and H are also constant in t.

However, this does not entail one to drop the Ṁ term appearing in eq.( 6), because such a time constancy can be used only after having derived the equations of motion. In deriving (6) the so called outer gauge has been used i.e. in which the radial function R(r)

appearing in the metric (1) is equal to r for r > r, and is deformed below r on a finite tract by a smooth function g of bounded support with g(x) = 0 for x ≥ 0, g lef t (0) = 1 and [14,17] R(r, t) = r + V (t) r g(rr(t)) (7) due to the fact that as a consequence of the constraints

∆R ≡ R (r + 0) -R (r -0) = - V R , V = p2 + m 2 (8)
being m the mass of the shell. The imposition of the constraints is the reason why one cannot adopt R = r for all r.

In the inner gauge a term of type Ḣ will appear while in a generic gauge both Ṁ and Ḣ terms will appear. Moreover, the constraints impose

∆π L = -p (9)
being

π L = R (R ) 2 -1 + 2M R ≡ RW (R, R , M)
where by M we denote the mass which is function constant in r for r > r and also for r < r but discontinuous at r.

The general form of p c is given by [17] p c = r(∆L -∆B)

with L and B given by eqs. (4,5). Using relations (8,9) p c becomes

p c (r) = √ 2M r - √ 2H r -r log r + √ p2 + m 2 -p - √ 2H r r - √ 2M r (10)
where [14,17] p is given implicitly by the solution of the equation

H -M = V + m 2 2r -p 2H r . (11)
We stress that no limit procedure is necessary to obtain (10,11) which hold with any deformation to the left of r.

In the inner gauges in which the deformation is taken to the right of r one obtains for the

canonical momentum p c p c = √ 2M r - √ 2H r -r log r - √ 2H r r -V i + pi - √ 2M r
where now pi is given by the implicit equation

H -M = V i - m 2 2r -pi 2M r , V i = pi + m 2
which is different from eq.( 11). More general gauges can also be considered in which the discontinuity ( 8) in R at r = r is partly due to a deformation on the right and to a deformation on the left of r. It is possible to show that actually p c is independent from any gauge choices in which L = 1. In fact after some algebra given in Appendix A, we find

p c = √ 2M r - √ 2H r -r log r - √ 2H r r - √ 2M r -r log r -H -M -m 2 2r - r√ C r -2H (12)
being C the discriminant

C = ( H -M r ) 2 + m 2 r2 ( H + M r -1) + m 4 4r 4 .
Notice that the argument of the second logarithm becomes 1 for m 2 = 0 and in this situation the last term in ( 12) disappears.

The mass m of the shell does not play a really important role in the phenomena we shall examine in the following, the reason being that, as seen from eq.( 43) of the Appendix at the horizon r = 2H where p c is singular p diverges and it cancels the term V and we know that Hawking radiation which is a late time phenomenon depends on the behavior of the modes at the horizon. Moreover eq.( 16) below holds also for m = 0.

For m = 0

p c = √ 2M r - √ 2H r -r log r - √ 2H r r - √ 2M r (13)
which is the original result by Kraus and Wilczek [1]. The space-part of the semiclassical mode is given by

u(r) = const exp(i( r r1 p c (r)dr + const)) ( 14
)
where r1 is any value of the radius outside the horizon and the additive constant can be taken also to depend on H, M and contributes only to a phase factor in the semiclassical mode. For m = 0 the integral appearing in eq.( 14) can be easily computed. One finds

p c (r)dr = f (r, M) -f (r, H) ( 1 5 )
where

f (r, M) = r2 -4M 2 2 log( √ r - √ 2M ) + r -2M 2 ( √ 2M r + r 2 ).
The large r behavior of p c is lim

r→∞ p c (r) = H -M = p2 (+∞) + m 2 ≡ ω.
The p c as a function of r develops an imaginary part iπr in the interval (2M, 2H). This is evident from eq.( 13) but in [17] it was shown to be true also in presence of mass i.e. for eq.( 12) independently of the value of m < H -M; the last is the necessary and sufficient condition for a massive shell to reach space infinity. One has as a consequence

Im rm r 0 p c dr = 2π(H 2 -M 2 ). ( 16
)
Thus we have shown that p c is independent of the gauge choice, within the family of the Painlevé gauges, and that eq.( 16) holds also for m = 0.

The exponential of minus eq.( 16) was given the interpretation of a tunneling amplitude by Parikh and Wilczek [3]. Criticism and alternative proposal for the tunneling amplitudes were given in the literature [4,5,6,7,8,9,10,11,12].

Here however, we shall not pursue this line of thought and instead still employing eqs. (10,11,15) we shall go back to the calculation of coefficients of the Bogoliubov transformation.

The expansion in ω = H -M of eq.( 15) to second order in ω is

ω M + 2 √ 2M r + r + 4M log( √ r - √ 2M √ 2M ) + ω 2 2 1 - 2 √ 2M √ r - √ 2M + √ 2r √ M + 4 log( √ r - √ 2M √ 2M ) + • • • (17)
What matters in the late time emission is the behavior of the mode at the horizon which is given by

4Mω log( r -2M 4M ) + 4Mω 2 1 2M log( r -2M 4M ) - 1 r -2M + • • •
We adopt the usual notation [1] for the expansion of the scalar field

φ φ = ∞ 0 dω √ 2ω (u ω (r)e -iωt a(ω) + u * ω (r)e iωt a + (ω))
where now t is the Painlevé time and u ω (r) is given by eq.( 14). The same field φ can be expanded in modes which are regular at the horizon

φ = ∞ 0 dk √ 2k (v k (r, t)b(k) + v * k (r, t)b + (k)).
The behavior of such v modes at the horizon is given by translating the expression [19]

v k = 1 √ 2π e ik(X-T ) with X = M 2 (V -U ) , T = M 2 (V + U ) , U = -e -u 4M , V = e v 4M
into the Painlevé coordinates, being

u = t s -r * , v = t s + r * , r * = r + 2M log( r 2M -1)
and t s the Schwarzschild time

t s = t -2 √ 2M r -2M log √ r - √ 2M √ r + √ 2M + const.
Near the horizon one finds

v k (r, t) = 1 √ 2π exp(ik(r -2M)e -t 4M ). ( 18
)
It is essential in obtaining eq.( 18) to adopt the Painlevé time. In fact such regular modes will be subsequently analyzed in terms of the u-mode which live also on the Painlevé background.

The usual technique for extracting the Bogoliubov coefficient [START_REF] Birrel | Quantum field theory in curved space[END_REF] is that of projection of the regular v-modes on the u-modes by space integration. Here, the main difference w.r.t.

the usual treatment is that the background metric is now the Painlevé metric with the mass M. Despite being the Painlevé metric a non static metric still it describes a static space time and thus due to the invariance of the space integrals the usual formalism for computing the scalar products works and there is no need to adopt the general form for stationary spaces [START_REF] Ashtekar | [END_REF]. Such a scalar product is given by

-i (ψ * 2 ∂ ρ ψ 1 -ψ 1 ∂ ρ ψ * 2 )g ρt √ -g ε trθφ drdθdφ (19)
where the integration extends to the region outside the horizon. Using √ -g = r2 sin θ we have for eq.( 19)

4πi ∞ 2M (ψ * 2 ∂ t ψ 1 -ψ 1 ∂ t ψ * 2 )r 2 dr -4πi ∞ 2M (ψ * 2 ∂ r ψ 1 -ψ 1 ∂ r ψ * 2 )N r r2 dr, N r = g rt = 2M r .
Taking into account that eq. (14,18) are reduced radial modes we find for the most singular term in the integrand giving the scalar product of v k with u ω √ M π ∞ 0 e -ikxτ e 4iωM log x-iωt dx x

where such term originates from g rt v * k ∂ r u ω and τ = e -t 4M , x = r -4M and we took into account the normalization of the u ω .

Integrating in x = r -2M we have

β * kω = √ M π ω k ∞ 0 e ikxτ e 4iM ω log x-iωt dx x = √ M π ω k ∞ 0
e ikxτ e 4iM ω log(xτ ) d(xτ ) xτ while α kω is obtained changing ω in -ω. On the other hand one can extract the Bogoliubov coefficients also performing a time Fourier transform of the regular modes at fixed r

β * kω = - 1 4 √ M π ω k ∞ -∞ e ikxτ e 4iM ω log x-iωt dt = √ M π ω k ∞ 0 e ikxτ e 4iM ω log(xτ ) d(xτ ) xτ
which also shows that the result does not depend on the value of r. The independence of the time Fourier transform from r can be proved on general grounds for the exact modes.

Thus, we have that the first order term reproduces the well-known Hawking integrals and the treatment outlined above is obviously equivalent to the original external field treatment even if we developed it in the Painlevé reference frame.

There are several reasons why one cannot exploit the second order term in the expansion (17). The second order contains a non integrable singularity in r; in addition it would not be proper to compute the scalar product of eq.( 17) with the regular mode v eq.( 18)

as also here one should consider the correction due to the back reaction. In addition one cannot employ any longer the Painlevè background metric characterized by M but one should also consider the corrections to it due to the presence of the shell. For this reasons in Sec. 4 we shall adopt the time Fourier transform technique as done in [1] 3

The equations of motion

We recall that in deriving the equations of motion one can vary H keeping M as a fixed datum or vary M keeping H as a fixed datum [17]. The first procedure is much simpler if one adopts the outer gauge because the term in Ṁ is zero in eq.( 6) and one obtains

ṙ ∂p c ∂H -N (r m ) = 0 ( 20
)
where N (r m ) does not depend on r m for r m > r and the usual normalization N (r m ) = 1 corresponds to identify t with the Killing time at space infinity. In this case eq.( 20)

becomes [14,17]

dr dt =   p V - 2H r   . (21)
The procedure in which one varies M keeping H fixed is more complicated due to the presence of the term in Ṁ in action (6) and which cannot be neglected to obtain the correct equations of motion. It was explicitely proven in [17] that one gets the same equations of motion [START_REF] Ashtekar | [END_REF] if one maintains the normalization N (r m ) = 1. On the other hand if one adopts the normalization N (r 0 ) = 1 one obtains the equations of motion w.r.t.

the time which flows at r 0 and which we shall denote by t . The calculation is most easily performed in the inner gauge obtaining

dr dt ∂p c ∂M + N (r 0 ) = 0 , with ∂p c ∂M =   pi V i - 2M r   -1 (22)
and thus

dr dt =   pi V i - 2M r   . (23)
Thus, we see that, due to the sign of the boundary term in eq.( 6), choosing N (r 0 ) = 1

gives -M as the Hamiltonian in this scheme.

We shall use both calculational schemes in the next section. The values of H and M which on shell are constant of motions give the values of the mass contained in r < r m and the mass contained in r < r 0 . Despite being the coordinate r one dimensional, r 0 and r m do not play an exactly symmetrical role as in deriving the action the Jacobian R 2 plays an essential role.

The late time expansion

In this section we give a simplified derivation of the late time expansion of the outgoing modes which are regular at the horizon. In constructing such regular modes one can use the scheme in which H plays the role of the Hamiltonian while M is a given fixed parameter. This is the scheme followed in [1,13]. One can also construct regular modes by giving H as a fixed parameter and using M as Hamiltonian as was described in Sec.

3 in the second derivation of the equations of motion.

We start from the case in which M is a given constant, while H plays the role of the Hamiltonian. The distinguishing feature of Hawking radiation is that of being a late time effect. Below we shall compute with a simple technique the late time expansion of the regular mode confining ourselves to the (ω/M) 2 corrections to the Hawking result. To that end it is sufficient to compute just the first order correction in the late time expansion.

In the expansion the terms are classified in power of τ = e -t 4M and powers of t. We shall need only the O(τ ) and O(tτ 2 ) terms which can be extracted in a very simple manner.

By keeping only the singular terms in p c and in the time development we have the equations which were used in [1]

p c = -r log √ r - √ 2H √ r - √ 2M (24) t = 4H log( √ r - √ 2H) -4H log( r(0) - √ 2H). ( 25
)
Furthermore, regularity is obtained by imposing that at t = 0 p c = k with k > 0. Thus we have the further restriction

0 < k = -r(0) log r(0) - √ 2H r(0) - √ 2M ( 26
)
The action is given by [1,13] kr(0)

+ t 0 (p c (r(t ), H(t), M, k) ṙ(t ) -H(t))dt (27)
where for once we wrote explicitely the dependence of p c on time. p c is given by the solution of the equation of motion with the two conditions p c = k at t = 0 and r(t) = r, being r a fixed value of the shell coordinate outside the horizon and t an arbitrary time.

In the procedure we shall develop below the only thing we shall need is the function H(t)

which we shall compute in the late time expansion. In the previous as in the following equations till eq.( 36) H stays for H(t).

From eq.( 25) we obtain r(0) -

√ 2H = ( √ r - √ 2H)e -t 4H
which substituted in eq.( 26) gives the implicit equation for the time evolution of

H(t) k = ( √ 2H + ( √ r - √ 2H)e -t 4H ) 2 log ( √ r - √ 2H)e -t 4H + δ H ( √ r - √ 2H)e -t 4H (28)
with

δ H = √ 2H - √ 2M. ( 29
)
From eq.( 28) we see that t going to +∞, √ 2H goes over to the finite value √ 2M because δ H has to go to zero. To lowest order we have

δ H = c H T with T ≡ e -t
4H which substituted in eq.( 28) gives for c

H c H = (e k 2M -1)L, L ≡ √ r - √ 2M .
Having determined the constant c H we can go over to the next term. By simply expanding eq.( 28) we obtain for large t t

4H(t) = t 4(M + √ 2M c H e -t 4M ) = t 4M 1 - √ 2M M c H τ + O(τ 2 ) (30) giving δ H ≡ √ 2H - √ 2M = c H τ + (c H τ ) 2 t (2M) 3/2 + O(τ 2 ) or H(t) = M + √ 2M c H τ + t 2M (c H τ ) 2 + O(τ 2 ) ( 3 1 )
where τ = e -t 4M . The last obtained relation ( 31) is necessary and sufficient to compute the ω 2 corrections in the late time framework. In fact, we are interested only in the time dependence of the mode given by the exponential of i times the expression (27) and we have [13]

∂S ∂t = -H(t)
which holds also with the boundary conditions (26) as can be explicitely verified. Thus the time development of S is given by

S = f (r) - t 0 H(t )dt .
Integrating we find

t 0 H(t )dt = const + Mt -4M √ 2M τ 1 -tτ 2 1 + ..
where for notational simplicity we set τ 1 = c H τ .

Thus, S behaves at large times as -Mt independently of k. On the other hand the Fourier time analysis of e iS contains frequencies which are above and below such value M and this is the well known fact that the mode of the system which is regular at the horizon does not represent an eigenvalue of the energy as measured by a stationary observer at space infinity. The deviations from the value M represent the positive and negative frequency content of the radiation mode. Thus, after subtracting the background frequency M we have in the time Fourier analysis the exponent

i(S -Mt -ωt) = i(f (r) - t 0 H(t )dt + Mt ± ωt). (32)
The saddle point -H(t) + M ± ω = 0 is given by

√ 2Mτ 1 + t 2M τ 2 1 ∓ ω = 0. ( 33
)
The upper sign refers to the calculation of the α coefficients while the lower sign refers to the β coefficients. For the upper sign we see that the saddle point is real thus giving in the saddle approximation the contribution 1 to the modulus of α. For the lower sign, i.e.

for the β coefficient, the exponent (32) becomes, using eq.( 33)

i[4M √ 2M τ 1 + t(τ 2 1 + ω)] = -i[4Mω + tω(1 + ω 2M )]. (34)
Thus to compute |β kω /α kω | to second order in ω we need Imt to first order in ω.

We have

√ 2M τ 1 = -ω(1 + t 4M 2 ω) = -ω(1 - ω M log(- ω √ 2Mc H ))
so that to first order ω we have

t = -4M(1 - ω M ) log(- ω √ 2M c H ) giving Imt = 4πM(1 - ω M )
Combining with eq.( 34)

β kω α kω = e -4πM (1-ω 2M ) . (35)
This is the explicit derivation through the late time expansion of the result obtained in [13] which corrects a previous result in [1]. From eq.( 35) through a standard procedure [1] which exploits the Wronskian property of the Bogoliubov coefficients one obtains the spectrum of the radiation

F (ω)dω = dω 2π 1 e 8πM ω(1-ω 2M ) -1 . (36)
In the above treatment the parameter M which characterizes the regular modes has to be identified with the semiclassical mass of the system which emits the radiation because it is the value of the Hamiltonian H(t) at large values of the time t and H(t) is the boundary term at r m which is related to the total energy of the system, i.e. the exterior mass.

In order to clarify the issue we want now to repeat the analysis from a different view point, i.e. by considering H as a given parameter and M, i.e. the interior mass, as Hamiltonian.

Actually from the boundary term in eq.( 6) with N (r 0 ) = 1 we see that the Hamiltonian i.e. the generator of the time translations is -M as we discussed at the end of Sec.

3. Thus we have to compute the regular mode of the system where H is a parameter with a given fixed value and the role of the Hamiltonian is played by -M. The physical identification of the parameter H is obtained after eq.( 41) from the time development of the regular mode. Taking into account the equation of motion (23) we have now

t = 4M log( √ r - √ 2M ) -4M log( r(0) - √ 2M ).
while p c has still the form (24). Furthermore, regularity is obtained by imposing that at t = 0, p c = k with k > 0. Thus we have the restriction

0 < k = -r(0) log r(0) - √ 2H r(0) - √ 2M
where now H is a constant parameter and M is a function of t . The implicit equation for M(t ) we obtain is

k = ( √ 2M + T ( √ r - √ 2M )) 2 log ( √ r - √ 2M)T ( √ r - √ 2M)T -δ M (37) with T = e -t 4M (t ) , and δ M = √ 2H - √ 2M.
For t → ∞ we have M(t ) → H and thus expanding in the implicit variable T we have

δ M = c M T + O(T 2 ) ( 3 8 )
where c M is easily computed from eq.( 37)

c M = (1 -e -k 2H )L , L = √ r - √ 2H.
Repeating the same procedure as the one described in eq.( 30) and squaring the relation (38) we obtain

M(t ) = H - √ 2Hc M τ + t 2H (c M τ ) 2
which is the analogue of eq.( 31). Then, the v k mode, i.e. the mode regular at the horizon is given by e i(kr(0)+ t 0 pc ṙ dt +M (t )t ) .

Again we have ∂(kr(0)

+ t 0 p c ṙ dt + M(t )t ) ∂t = M(t )
so that the time dependence of the regular mode for fixed r is

e i(f (r)+ t 0 M (t )dt ) .
For t → +∞ we have M(t ) → H. Subtracting such background frequency we have the radiation mode regular at the horizon e i(f (r)+ t 0 M (t )dt -Ht ) .

We which is the same as eq.( 32) with -i t 0 H(t )dt + iMt replaced by -iHt + i t 0 M(t )dt . Repeating the steps after eq.( 32) we obtain for the saddle point referring to the calculation of the β coefficient the relation

H -M(t ) = √ 2Hτ 1 - t 2H (τ 1 ) 2 = -ω
and for the saddle point value of the exponent

-i[4Hω + t ω(1 - ω 2H )]. (39)
Again we need t to first order in ω.

t = -4M(1 + ω H ) log(- ω √ 2Hc M ) , Imt = -4M(1 + ω H ). (40)
Combining eq.( 39) and (40) we obtain the ratio

β kω α kω = e -4πωH(1+ ω 2H ) .
Now the asymptotic behavior of the time development of the regular mode in the t representation is e iHt and being t related to the interior mass we have to identify the parameter H as the mass of the remnant of the black hole after the emission of the quantum of energy ω. This is in agreement with eq.( 35) where as stated at the beginning of the present section we have been working consistently to second order in ω/M.

The validity of the late time expansion can be examined by comparing the first order result, in eq.( 34) with the second order term. The typical frequency of the Hawking radiation is ω ≈ l 2 P /M with M the mass of the black hole and l P the Planck length; at such a frequency the ratio of the second order to the first is l 2 P /M 2 apart logarithmic corrections. Thus the expansion holds for black holes with mass of a few Planck masses or higher mass.

Conclusions

In this paper we have examined some issues related to the semiclassical treatment of Hawking radiation. The main advantage of this approach is that it allows one to treat the backreaction, i.e. the fact that during the transition the mass of the black hole varies.

The reduced Hamiltonian description is a critical component of this approach and such a reduction can be performed rigorously at the classical level. The introduction of the generating function F in Sec. 2 greatly simplifies the reduction process. We proved that the canonical conjugate momentum p c which appears in the reduced action and which plays the major role in all subsequent developments, is invariant in the class of the Painlevé gauges. We do not expect this to be true for all possible gauges but the Painlevé class of gauges, which are defined by the choice L ≡ 1 in the metric (1), gives a good description of an asymptotic observer at space infinity and has the advantage of being nonsingular at the horizon. There is still some gauge freedom in the choice of R but we proved that p c does not depend on this choice. The classical dynamics can be developed using either the exterior mass or the interior mass as Hamiltonian and these two descriptions are equivalent.

By interpreting the exponential of the reduced semiclassical action as modes of the system, as was done in [1], it is possible to calculate the Bogoliubov coefficients. We gave a simplified treatment of the late time expansion introduced in [1] leading to the backreaction corrections, both using the exterior and the interior mass as Hamiltonians. The late time expansion used in this treatment is expected to hold for black holes whose mass is a few Planck masses or more. In [17] formula (16) was proven also for the emission of two interacting shells (massive or massless) which can cross during their time evolution. This would point to the absence of correlations among emitted quanta [START_REF] Parikh | Rio de Janeiro 2003, Recent developments in theoretical and experimental general relativity, gravitation, and relativistic field theories[END_REF]. However, up to now we have no mode interpretation of such a result. eq.( 41) and multiplying both numerator and denominator by R + + W + we obtain for p c

p c R = 2M R - 2H R + log 1 -2M R 1 + 2H R 1 -2H R + H-M R -m 2 2R 2 - √ C = = 2M R - 2H R + log 1 -2H R + H-M R -m 2 2R 2 + √ C 1 + 2M R 1 -2H R
showing the gauge independence of p c within the family of Painlevè gauges.

For the discussion of the analytic properties of p c it is however simpler to use the system eq. (10,11) as done in [17].

2 .∞ dt e -iHt +i t 0 M

 20 have now to analyze such regular mode in terms of the u-modes which are characterized as in the previous treatment by well defined values of M and H. On dealing with -M as Hamiltonian one can use either the outer or the inner gauge with no change in the results in p c as we have stressed in Sec. 2. By subtracting again the background frequency Ht from the total mode e ) with ω ≡ H -M = p2 (+∞) + m Thus we have for the time Fourier analysis at fixed r of the regular mode in terms of the u-modes +∞ -(t )dt e ±iωt

Appendix A

In this appendix we derive the explicit expression for the conjugate canonical momentum p c in the class of Painlevè gauges, showing its universality within these gauges. The general form of p c was given in [17]

where R stays for R(r) which in our scheme equals r, R + and R -stay for the right and left derivatives of R at r. We solve in terms of R + using the two relations (8,9)

By squaring the second equation we obtain

Squaring again we obtain for p the second oder equation

whose discriminant is given by

from which we see that p is a gauge dependent quantity i.e. a quantity which depends on R + . Using eq.( 42) with p given by the previous equation substituting R -and W -into