On the Rankin-Selberg method for higher genus string amplitudes

0.91 MB·

by  Ioannis Florakis

CERN-TH-2016-022
arXiv:1602.00308v2
On the Rankin-Selberg method
6
for higher genus string amplitudes
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2

b
e
F
Ioannis Florakis1 and Boris Pioline1,2
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]
h 1 CERN Theory Division, 1211 Geneva 23, Switzerland
t
- 2 Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR
p
7589,
e
h Université Pierre et Marie Curie - Paris 6, 4 place Jussieu, 75252
[ Paris cedex 05, France

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v
8
Abstract
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3
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Closed string amplitudes at genus h ≤ 3 are given by integrals of Siegel modular functions
0
. on a fundamental domain of the Siegel upper half-plane. When the integrand is of rapid
2
decay near the cusps, the integral can be computed by the Rankin-Selberg method, which
0
6 consists of inserting an Eisenstein series E (s) in the integrand, computing the integral by the
h
1
orbit method, and finally extracting the residue at a suitable value of s. String amplitudes,
:
v however,typicallyinvolveintegrandswithpolynomialorevenexponentialgrowthatthecusps,
i
X and a renormalization scheme is required to treat infrared divergences. Generalizing Zagier’s
r extension of the Rankin-Selberg method at genus one, we develop the Rankin-Selberg method
a
for Siegel modular functions of degree 2 and 3 with polynomial growth near the cusps. In
particular, we show that the renormalized modular integral of the Siegel-Narain partition
function of an even self-dual lattice of signature (d,d) is proportional to a residue of the
Langlands-Eisenstein series attached to the h-th antisymmetric tensor representation of the
T-duality group O(d,d,Z).
E-mail: [email protected]
[email protected]
Contents
1 Introduction 1
2 Degree one 6
3 Degree two 10
3.1 Regularizing the divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Renormalizing the integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Constructing ♦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2
3.4 Lattice partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Product of two Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Degree three 25
4.1 Regularizing the divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Renormalizing the integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Constructing ♦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3
4.4 Lattice partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A Siegel-Eisenstein series and invariant differential operators 42
B Lattice partition function and Langlands-Eisenstein series 46
C Laplace equation at genus 3 48
1 Introduction
According to the current rules of perturbative superstring theory, scattering amplitudes at
h-loops are expressed as integrals of a suitable superconformal correlation function on the
moduli space M of super-Riemann surfaces of genus h [1,2]. Although there is in general no
h
global holomorphic projection onto the moduli space M of ordinary Riemann surfaces [3],
h
for most practical purposes the integral over M can be reduced to an integral over M ,
h h
possibly supplemented with boundary contributions from nodal curves, which incorporate the
infrared singularities due to massless degrees of freedom. Even after this reduction has been
performed, thereareveryfewcaseswheretheintegraloverM canbeevaluatedexplicity, due
h
tothecomplexityoftheintegrandandoftheintegrationdomain,aquotientoftheTeichmüller
space T by the mapping class group Γ . For h ≤ 3, T is isomorphic (via the period map
h h h
Σ → Ω) to the Siegel upper half plane H of degree h (away from the separating degeneration
h
locus), while Γ is isomorphic to the Siegel modular group PSp(2h,Z) = Sp(2h,Z)/{±1} (or
h
a congruence subgroup thereof, if one keeps track of spin structures).
At genus one, several efficient methods for integrating modular functions on the funda-
mental domain F of the Poincaré upper-half plane have been developped, starting with the
1
integration-by-partsmethod[4]andtheorbitmethodof[5–8]inthephysicsliterature,andthe
Rankin-Selberg-Zagier method for automorphic functions of polynomial growth in the math
1
literature [9]. With the recent advances of [10–13]1, it is now possible to evaluate analytically
the integral over F of any (modular invariant) product of an almost weakly holomorphic
1
modular form times a lattice partition function. This covers most of the cases relevant for
BPS-saturated amplitudes at one-loop.
At genus two or three, the only examples computed so far are those where the integrand is
constant [20,21], or proportional to the so-called Kawazumi-Zhang invariant ϕ (Ω) [22,23],
KZ
or proportional to the Narain partition function Γ of the even self-dual lattice of signature
d,d,h
(d,d)[24,25]. Inthefirstcase,thevolumeofthefundamentaldomainF oftheSiegelmodular
h
groupofdegreehiswell-knownsince[26],whileinthesecondcasetheintegralofϕ couldbe
KZ
(cid:82)
computed by integration by parts [23]. In the last case, the integral dµ Γ is divergent
F h d,d,h
h
ford ≥ h+1andmustberegularized. Itwasconjecturedin[24]thattherenormalizedintegral
(cid:63),SO(d,d)
shouldbeproportionaltothesumoftheLanglands-EisensteinseriesE (s = h)attached
S,C
to the spinorial representations of the T-duality group SO(d,d,Z).
(cid:82)
In[27], somepreliminarystepsweretakentowardsevaluatingtheintegral dµ Γ by
F h d,d,h
h
the Rankin-Selberg method. Recall that for a modular function F of rapid decay, the integral
(cid:82)
dµ F(Ω) can be deduced from the ‘Rankin-Selberg transform’
F h
h
(cid:90)
R(cid:63)(F,s) = dµ E(cid:63)(s,Ω)F(Ω) , (1.1)
h h h
F
h
whereE(cid:63)(s,Ω)isthe(completed)non-holomorphicSiegel-Eisensteinseries,byusingtheknown
h
fact [28] that E(cid:63)(s,Ω) is a meromorphic function of s ∈ C with a pole at s = h+1, and with
2
constant residue r = 1 (cid:81)(cid:98)h/2(cid:99) ζ(cid:63)(2j +1) :
h 2 j=1
(cid:90)
1
dµ F(Ω) = Res R(cid:63)(F,s) . (1.2)
h r s=h+1 h
F h 2
h
The Rankin-Selberg transform (1.1) can in turn be computed by representing the Eisenstein
series E(cid:63)(s,Ω) for Re(s) > h+1 as a sum over images under Γ ,
h 2 h
(cid:98)h/2(cid:99) (cid:32) (cid:33)
(cid:89) (cid:88) A B
E(cid:63)(s,Ω) = ζ(cid:63)(2s) ζ(cid:63)(4s−2j) |Ω |s|γ , Γ = Γ ∩{ }, (1.3)
h 2 h,∞ h 0 D
j=1 γ∈Γ \Γ
h,∞ h
where Ω = Ω +iΩ , |Ω | ≡ detΩ , and unfolding the integration domain against the sum
1 2 2 2
over images, leading to
(cid:98)h/2(cid:99) (cid:90)
(cid:89)
R(cid:63)(F,s) = ζ(cid:63)(2s) ζ(cid:63)(4s−2j) dµ |Ω |sF(Ω) . (1.4)
h h 2
Γ \H
j=1 ∞,h h
The integral over Ω projects F(Ω) onto the zero-th Fourier coefficient F(0)(Ω ) with respect
1 2
to Γ , while the remaining integral over Ω runs over a fundamental domain of the action
h,∞ 2
1OtherattemptstousetheRankin-Selbergmethodtocomputestringamplitudesorprobetheasymptotic
density of string states include [14–19].
2
of PGL(h,Z) on the space P of positive definite symmetric matrices. The latter can be
h
viewed as a generalized Mellin transform of F(0)(Ω ). Notably it inherits the analyticity
2
property and invariance under s (cid:55)→ h+1 − s of the Eisenstein series E(cid:63)(s,Ω). In the case
2 h
where F(Ω) = |Ω |w|Ψ(Ω)|2, where Ψ(Ω) is a holomorphic cusp form of weight w, the integral
2
over P /PGL(h,Z) can be computed using again the unfolding method and identity (B.2) of
h
appendix§B.ThisexpressesR(cid:63)(F,s)asanL-seriesgeneralizingthesymmetricsquareL-series
h
for h = 1 [29–32], establishing its analytic properties and functional equation.
If however F(Ω) does not decay sufficiently rapidly at the cusps, as is typically the case in
string amplitudes, the integrals (1.1) and (1.4) must be regularized. For h = 1, it was shown
in [9] that for modular functions with polynomial growth at the cusp, a suitably renormal-
ized version of the integral (1.1) is given by the Mellin transform of F(0) −ϕ (still denoted
by R(cid:63)(F,s)), where ϕ(Ω ) is the non-decaying part of F(Ω); the latter has a meromorphic
h 2
continuation in s with additional poles over and above those of the Eisenstein series E(cid:63)(s,Ω).
h
Moreover,therenormalizedintegralofF differsfrom(oneoverr times)theresidueofR(cid:63)(F,s)
h h
at s = h+1 by a computable term whenever the order of the pole is greater than one. In the
2
(cid:82)
context of string theory, the regularization of a physical amplitude dµ F(Ω) by inserting
F h
h
anEisensteinseriesintheintegrandandthenextractingtheresidueats = h+1,canbeviewed
2
as an analogue of dimensional regularization in quantum field theory, where the number of
non-compact space-time dimensions is analytically continued from D to D−(2s−h−1).
Assuming that a similar procedure could be carried out for h > 1, it was found in [27] that
fortheparticularcaseF = Γ , therenormalizedRankin-Selbergtransform(1.4)(aftersub-
d,d,h
tracting by hand all non-decaying terms from F) is proportional to the Langlands-Eisenstein
series E(cid:63),SO(d,d)(s+d−1−h) attached to the h-th antisymmetric power of the fundamental rep-
ΛhV 2
resentation; and that the renormalized integral of F is equal to the residue of the same2 at
s = h+1, up to an undetermined correction term δ when the order of the pole is greater than
2
one. The arguments in [27] were heuristic, however, and one motivation for the present work
is to put the claim of [27] on solid footing.
More generally, the goal of this work is to extend the Rankin-Selberg-Zagier method [9]
to Siegel modular functions of degree h ≥ 2 which are regular inside H and have at most
h
polynomial growth at the cusps. The main challenge is to find a convenient cut-off which
removes all divergences, while allowing to unfold the integration domain against the sum
in the Siegel-Eisenstein series. In general, divergences originate from regions of F where
h
a diagonal block of size h × h in the lower-right corner of Ω is scaled to infinity, while
2 2 2
keeping the remaining entries in Ω finite. The stabilizer of this cusp inside PSp(2h,Z) is a
parabolic subgroup of the Siegel modular group with Levi component equal to [Sp(2h ,Z)×
1
GL(h ,Z)]/Z , where h +h = h. In the language of string theory (so for h = 2,3 only), this
2 2 1 2
divergence is interpreted as a h -loop infrared subdivergence for 1 ≤ h < h, or as a primitive
2 2
infrared divergence for h = h. To subtract these divergences, we shall apply the following
2
strategy (already suggested in [10]):
i) construct an increasing family of compact domains FΛ ⊂ F , Λ ∈ R+ such that
h h
2This is compatible with the fact that the same integral is proportional to the residue of E(cid:63),SO(d,d)(s) at
S,C
s=h, thanks to identities between Langlands-Eisenstein series.
3
lim FΛ = F ; the regularized integral R(cid:63),Λ(F,s) = (cid:82) dµ E(cid:63)(s)F on the ‘trun-
Λ→∞ h h h FΛ h h
h
catedfundamentaldomain’FΛ isthenmanifestlyfiniteandhasthesameanalyticstruc-
h
ture as E(cid:63)(s) as a function of s ∈ C;
h
ii) find an invariant differential operator ♦ which annihilates the non-decaying part of the
integrand F;
iii) relate the regularized integrals R(cid:63),Λ(F,s) and R(cid:63),Λ(♦F,s) using integration by parts;
h h
iv) apply the standard Rankin-Selberg method to the decaying function ♦F, i.e. compute
lim R(cid:63),Λ(♦F,s) in terms of the (generalized) Mellin transform of the constant term
Λ→∞ h
♦F(0);
v) relate the Mellin transform of ♦F(0) to the regularized Mellin transform of F(0).
In the body of this paper, we develop this strategy in detail in degree one (revisiting [9,10]),
degreetwo, anddegreethree. Inappendix§A,wecollectrelevantfactsaboutSiegel-Eisenstein
seriesandinvariantdifferentialoperatorsvalidinanydegree,whichcouldbeusedtoextendour
procedurebeyonddegreethree. Inappendix§B,wecomputetherenormalizedRankin-Selberg
transform for the Siegel-Narain lattice partition function in arbitrary degree. Aside from this
example, it would be interesting to use our techniques to study the analytic properties of
L-series associated to non-cuspidal Siegel modular forms.
Since we only consider integrals of Siegel modular functions which are regular inside H ,
h
our procedure only applies to string amplitudes of genus h ≤ 3, whose integrand is regular
in all separating degeneration limits, as well as in non-separating degenerations which do not
correspond to cusps of the Siegel upper half-plane (see Figure 1). Still, it is applicable to
a number of amplitudes of physical interest, such as the two-loop D4R4 [20] and three-loop
D6R4 [21]couplingsintypeIIstringtheorycompactifiedonTd, whichareproportionaltothe
renormalized modular integral of the lattice partition function Γ for h = 2 and h = 3, re-
d,d,h
spectively. Usingthetechniquesdevelopedinthepresentpaper,weestablishthatthetwo-loop
D4R4 and the three-loop D6R4 amplitudes are given by residues of the Langlands-Eisenstein
series E(cid:63),SO(d,d)(s(cid:48)) and E(cid:63),SO(d,d)(s(cid:48)) at s(cid:48) = d/2, respectively. Moreover, in appendix §C,
Λ2V Λ3V
using similar techniques as in [33], we show that the three-loop D6R4 amplitude satisfies a
Laplace-type differential equation as a function of the moduli of the torus Td, with anoma-
lous source terms originating from boundaries of the moduli space; the coefficients of these
anomalous terms agree perfectly with those predicted from S-duality [34].
An important challenge is to extend our techniques to Siegel modular forms with singu-
larities inside the Siegel upper half-plane. The special case of the modular integral of the
genus-two Kawazumi-Zhang invariant ϕ (Ω), relevant for two-loop D6R4 amplitudes in type
KZ
II string theory on tori, was considered in [33], leading to a novel construction of ϕ (Ω) [35].
KZ
It would be interesting to consider other examples with more severe singularities on the sepa-
rating degeneration locus, such as D2H4 amplitudes in type II string theory compactified on
K [36].
3
4
6
5
4
1
3
1 1
2 1 1
1
1
1 2
0
2 3
Codim. h=2 h=3
Figure 1: Non-separating degenerations of Riemann surfaces of genus two and three (see e.g.
Figures 1 and 4 in [37] for the full set including separating degenerations). The boxed ones
correspond to those where the period matrix reaches a boundary of the fundamental domain
F in the Siegel upper half-plane.
h
5
Acknowledgements. B.P. is grateful to Rodolfo Russo for collaboration on the related
work [33] which paved the way for this project. I.F. would like to thank the ICTP Trieste for
its kind hospitality during the final stages of this work.
2 Degree one
Asawarm-upforthehigherdegreecasesdiscussedinthiswork, letusrecoverthemainresults
of[9]usingthemethodoutlinedabove. LetF(τ)beanautomorphicfunctionofSL(2,Z)with
polynomial growth3 at the cusp τ = i∞,
(cid:96)
(cid:88)
F(τ) = ϕ(τ )+O(τ−N) ∀N > 0 , ϕ(τ ) = c τσi , (2.1)
2 2 2 i 2
i=1
where c ∈ C,σ ∈ C. We define the regularized Rankin-Selberg transform of F by
i i
(cid:90)
R(cid:63),Λ(F,s) ≡ dµ E(cid:63)(s,τ)F(τ), (2.2)
1 1 1
FΛ
1
where
1 (cid:88) τs
E(cid:63)(s,τ) ≡ ζ(cid:63)(2s) 2 , (2.3)
1 2 |cτ +d|2s
(c,d)∈Z2,
(c,d)=1
is the completed non-holomorphic Eisenstein series of weight 0, FΛ = {τ ∈ H ,|τ| > 1,−1 ≤
1 1 2
τ < 1,τ < Λ} is the ‘truncated fundamental domain’, and dµ = dτ dτ /τ2 (normalized
1 2 2 1 1 2 2
such that the volume of the fundamental domain is V = lim (cid:82) dµ = 2ζ(cid:63)(2) = π).
1 Λ→∞ FΛ 1 3
1
The non-decaying part ϕ is annihilated by the operator
(cid:96)
(cid:89)
♦ = (∆−σ (σ −1)) . (2.4)
i i
i=1
Using the Chowla-Selberg formula
E(cid:63)(s,τ) = ζ(cid:63)(2s)τs+ζ(cid:63)(2s−1)τ1−s+... , (2.5)
1 2 2
where the dots denote exponentially suppressed terms as τ → ∞, and integrating by parts (cid:96)
2
3A more general growth condition ϕ(τ2) = (cid:80)(cid:96)i=1ciτ2σi(logτ2)ni/ni! was considered in [9]. For simplicity
we shall assume n = 0. The case n > 0 can be dealt with by raising the i-th factor in (2.4) to the power
i i
n . We could also allow the coefficients c to be arbitrary periodic functions of τ . The subsequent analysis is
i i 1
(cid:82)1
unchanged provided c is understood to represent 2 c (τ )dτ .
i −1 i 1 1
2
6
times, one finds that the regularized Rankin-Selberg transform of ♦F is given by
(cid:96)
R(cid:63),Λ(♦F,s) =(cid:89)[s(s−1)−σ (σ −1)] R(cid:63),Λ(F,s)
1 i i 1
i=1

(cid:96)
(cid:88) (cid:89) (cid:89)
+ c ζ(cid:63)(2s) (s−σ ) (s−1+σ )Λσj+s−1
j i i (2.6)
j=1 i=1...(cid:96) i=1...(cid:96)
i(cid:54)=j

(cid:89) (cid:89)
−ζ(cid:63)(2s−1) (s−σ ) (s−1+σ )Λσj−s+...
i i 
i=1...(cid:96) i=1...(cid:96)
i(cid:54)=j
where the dots denote exponentially suppressed terms as Λ → ∞. Thus, reorganizing terms
and assuming that s does not coincide with any of the σ or 1−σ ,
i i
R(cid:63),Λ(F,s) =R(cid:63)1,Λ(♦F,s) +(cid:88)(cid:96) c (cid:18)ζ(cid:63)(2s)Λσj+s−1 + ζ(cid:63)(2s−1)Λσj−s(cid:19)+... (2.7)
1 D (s) j σ +s−1 σ −s
1 j j
j=1
where
(cid:96)
(cid:89)
D (s) = [s(s−1)−σ (σ −1)] . (2.8)
1 i i
i=1
On the other hand, since ♦F is decaying faster than any power of τ , the standard Rankin-
2
Selberg method shows that R(cid:63),Λ,s(♦F) has a finite limit at Λ → ∞, given by
1
(cid:90) ∞ (cid:90) 1/2
R(cid:63)(♦F,s) = ζ(cid:63)(2s) dτ τs−2 dτ ♦F , (2.9)
1 2 2 1
0 −1/2
and that R(cid:63),Λ(♦F) has a meromorphic continuation in the s plane, invariant under s (cid:55)→ 1−s,
1
with only simple poles at s = 0 and s = 1, due to the poles of E(cid:63)(s). Using the fact that
1
♦ϕ = 0 and integrating by parts, we have
(cid:90) ∞
R(cid:63)(♦F,s) = ζ(cid:63)(2s)D (s) dτ τs−2(F(0)−ϕ), (2.10)
1 1 2 2
0
where F(0)(τ) = (cid:82)1/2 dτ F. Thus, if we define the renormalized integral of E(cid:63)(s) times F by
−1/2 1 1
subtracting the divergent terms in (2.7),
(cid:90)
R(cid:63)(F,s) ≡R.N. dµ E(cid:63)(s)F
1 1 1
F1
  (2.11)
≡ Λl→im∞R(cid:63)1,Λ(F)−(cid:88)(cid:96) cj(cid:18)ζ(cid:63)(σ2js+)Λsσ−j+1s−1 + ζ(cid:63)(2sσ−j −1)sΛσj−s(cid:19) ,
j=1
7
then it follows from (2.7) and (2.9) that
(cid:90) ∞ R(cid:63)(♦F,s)
R(cid:63)(F,s) = ζ(cid:63)(2s) dτ τs−2(F(0)−ϕ) = 1 . (2.12)
1 2 2 D (s)
0 1
Thus the renormalized integral R(cid:63)(F,s) is equal to the Mellin transform of the subtracted
1
constant term F(0)−ϕ, and has a meromorphic continuation to the s plane, invariant under
s (cid:55)→ 1−s, with only simple poles at s ∈ {0,1,σ ,1−σ ,i = 1...(cid:96)} (assuming for now that
i i
no σ is equal to 1). For σ ∈/ {0,1}, the residue at s = σ originates from the subtraction in
i
(2.11),
(cid:96) (cid:96)
(cid:88) (cid:88)
Res R(cid:63)(F,s) = ζ(cid:63)(2σ −1)c − ζ(cid:63)(2σ −1)c . (2.13)
s=σ 1 i i i i
i=1 i=1
σ=σi σ=1−σi
Since the residue of E(cid:63)(s) at s = 1 is a constant r = Res ζ(cid:63)(2s−1) = 1, it is natural to
1 1 s=1 2
define the renormalized integral of F as twice the residue of R(cid:63)(F,s) at that point,
1
 
R.N.(cid:90)F1dµ1F = r11Ress=1R(cid:63)1(F,s) = Λl→im∞(cid:90)F1Λdµ1F −(cid:88)j=(cid:96)1 cσjjΛ−σj−11 . (2.14)
If, however, one of the σ ’s coincides with 1, then R(cid:63)(F,s) has a double pole at s = 0 and
i 1
s = 1, with coefficient 1 (cid:80) c . We may then define the renormalized integral as
2 σj=1 j
 
(cid:90) (cid:90) (cid:88) cjΛσj−1 (cid:88) 
R.N. dµ1F = lim  dµ1F − − cjlogΛ , (2.15)
F1 Λ→∞ F1Λ j=1...(cid:96) σj −1 j=1...(cid:96) 
σj(cid:54)=1 σj=1
in which case it differs from twice the residue of R(cid:63)(F,s) at s = 1 by an additive constant,
1
(cid:90)
(cid:88) π (cid:88)
R.N. dµ F = 2Res R(cid:63)(F,s)+ c log(4πe−γ)+ c . (2.16)
1 s=1 1 j 3 j
F1 j j
σj=1 σj=0
Asaprimeexampleofaone-loopmodularintegralrelevantforstringtheory,letusconsider
the case F = Γ , where Γ is the Siegel-Narain partition function for the even self-dual
d,d,1 d,d,h
lattice of signature (d,d), which we define here for arbitrary genus h (Ω = τ for h = 1):
Γd,d,h(g,B;Ω) = |Ω2|d/2 (cid:88) e−πLIJΩ2,IJ+2πimIini,JΩ1,IJ , (2.17)
(mI,ni,I)∈Z2dh
i
where
LIJ = (mI +B nI,k)gij(mJ +B nJ,l)+ni,Ig nj,J . (2.18)
i ik j jl ij
8
At genus one, L11 ≡ M2 is the squared mass of a closed string with momentum m1 and
i
winding number ni,1 along a d-dimensional torus with metric g and Kalb-Ramond field B .
ij ij
The positive definite matrix g and antisymmetric matrix B can be viewed as coordinates
ij ij
on the Grassmannian G = O(d,d)/[O(d)×O(d)], which parametrizes even self-dual lattices
d,d
d/2
of signature (d,d). In this case, ϕ(τ ) = τ , and the renormalized integral (2.12) is given by
2 2
ζ(cid:63)(2s)Γ(s+ d −1) (cid:88)
R(cid:63)(Γ ,s) = 2 M−2s−d+2. (2.19)
1 d,d,1 d
πs+2−1 (mi,ni)∈Z2d
mini=0,(mi,ni)(cid:54)=0
This is recognized as the Langlands-Eisenstein series of SO(d,d,Z) attached to the vector
representation,
R(cid:63)(F,s) = 2E(cid:63),SO(d,d)(s+ d −1) . (2.20)
1 V 2
(cid:63),SO(d,d)
In particular, it follows from the above that E (s) has a meromorphic continuation to
V
the s plane, invariant under s (cid:55)→ d−1−s, with simple poles (for d (cid:54)= 2) at s = 0, d−1, d,d−1
2 2
and residues
1
Res E(cid:63),SO(d,d)(s) = ζ(cid:63)(d−1), (2.21)
s=d−1 V 2
(cid:90)
1
(cid:63),SO(d,d)
Res E (s) = R.N. dµ Γ , (2.22)
s=d V 4 1 d,d,1
2 F1
(cid:82)
where we define the renormalized integral R.N. dµ Γ , for all values of d, as
F1 1 d,d,1
  
d
(cid:90) (cid:90) Λ2−1
R.N. F1dµ1Γd,d,1 = Λl→im∞ F1Λdµ1Γd,d,1−d2 −1Θ(d−2)+δd,2 logΛ . (2.23)
Here, Θ(x) = 1 if x > 0 and 0 otherwise. This analytic structure is consistent with the one
derived from the Langlands constant term formula (see e.g. Appendix A in [34]).
(cid:63),SO(2,2)
For d = 2, E (s) has a double pole at s = 1,
V
E(cid:63),SO(2,2)(s) = 1 + 1 (cid:2)2γ −log(cid:0)16π2T U |η(T)η(U)|4(cid:1)(cid:3) , (2.24)
V 4(s−1)2 4(s−1) 2 2
where we used E(cid:63),SO(2,2)(s) = E(cid:63)(s,T)E(cid:63)(s,U), [10] and the Kronecker limit formula
V 1 1
E(cid:63)(s,τ) = 1 − 1 log(cid:2)4πτ |η(τ)|4e−γ(cid:3)+O(s−1) . (2.25)
1 2(s−1) 2 2
The residue at the pole differs by an additive constant from (1/4 times) the renormalized
9