Computation of the spectral density of two-point functions: Complex masses,cut rules, and beyond

David Dudal1,* and Marcelo S. Guimaraes2,†

1Department of Physics and Astronomy, Ghent University, Krijgslaan 281-S9, 9000 Gent, Belgium2Departamento de Fısica Teorica, Instituto de Fısica, UERJ - Universidade do Estado do Rio de Janeiro,

Rua Sao Francisco Xavier 524, 20550-013 Maracana, Rio de Janeiro, Brazil(Received 8 December 2010; published 16 February 2011)

We present a steepest descent calculation of the Kallen-Lehmann spectral density of two-point

functions involving complex conjugate masses in Euclidean space. This problem occurs in studies of

(gauge) theories with Gribov-like propagators. As the presence of complex masses and the use of

Euclidean space brings the theory outside of the strict validity of the Cutkosky cut rules, we discuss an

alternative method based on the Widder inversion operator of the Stieltjes transformation. It turns out that

the results coincide with those obtained by naively applying the cut rules. We also point out the potential

usefulness of the Stieltjes (inversion) formalism when nonstandard propagators are used, in which case cut

rules are not available at all.

DOI: 10.1103/PhysRevD.83.045013 PACS numbers: 11.55.Fv, 11.55.Bq

I. INTRODUCTION

In the setting of local quantum field theory, all informa-tion on the spectrum resides in the Green functions of thetheory, as they build up the scattering S matrix. A specialclass of Green functions are the two-point functions. Weshould not only consider the elementary propagators of thetheory, but also the two-point functions of local compositeoperators, which correspond to propagating bound states,are relevant.

Scattering amplitudes are in general analytic functionsof the Lorentz invariants constructed from the externalmomenta characterizing a given process. In the presentwork we shall only be interested in two-point functionswhich can be considered as functions of the external mo-mentum k2 ¼ s, analytically continued to the complex splane with a cut on the real axis starting at s ¼ �0 � 0,which is identified as the threshold for multiparticle pro-duction. Such a two-point correlation function can be castin the form of a dispersion relation, that is, an integralrepresentation written in terms of the function’s disconti-nuities in the complex plane. For the two-point function ofan arbitrary operatorOwe can write the general expression

hOðkÞOð�kÞi ¼ F ðsÞ ¼ 1

�

Z 1

0d�

ImF ð�Þ�� s

: (1.1)

The imaginary part in the argument of the integral standsfor the discontinuity of the function F ðsÞ across the cut.This ought to be a positive definite quantity proportional tothe total cross section, as demanded by the optical theorem.The representation (1.1) is known as the Kallen-Lehmannspectral representation. For a concise treatment, let us referto e.g. [1–4].

This Kallen-Lehmann spectral representation can beconstructed, for the case of real masses in Minkowskispace, through the use of the Cutkosky cut rules [2,5],which state that, in order to calculate the discontinuityassociated with a Feynman diagram, we just need to cutpropagator lines in the diagram in all possible ways,thereby replacing the corresponding propagators by deltafunctions localizing to the physical phase space, and thento sum over all these contributions. Physically, we placethe particles running in these lines on-shell.Specifying for a moment to the strong interaction, it is

well known that QCD exhibits confinement; i.e. the color-charged elementary degrees of freedom (quarks and glu-ons) do not propagate in the physical world. The observ-ables are the color-neutral bound states, corresponding tothe gauge invariant operators constructed from the gluonand quarks fields, being the mesons, baryons, glueballs,and quark-gluon hybrids. A big contemporary challenge isto extract information on that nonperturbative spectrumusing theoretical, phenomenological, and/or numerical lat-tice tools. In the so-called sum rules approach to the QCDspectrum, spectral representations play a pivotal role; werefer to the comprehensive tome [6], to [7,8] for the semi-nal works, or to [4] for a pedestrian’s introduction.In many approaches to QCD, including the sum rules, an

important role is played by the elementary propagators, inparticular, the gluon propagator, which is usually studied inthe Landau gauge, @�A� ¼ 0. The elementary propagators

are the fundamental building blocks of more complicatedGreen functions, including the ones corresponding tobound states [9–11]. For some recent advancements onthe gluon propagator, including numerical lattice results,let us refer to the nonextensive list of [12–28].It is apparent that the Landau gauge gluon propagator

exhibits a violation of positivity [29,30], a fact which alsoreceived direct lattice confirmation [31,32]. This violation

*david.dudal@ugent.be†msguimaraes@uerj.br

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of positivity means nothing more than that there cannotexist a Kallen-Lehmann representation [11]; hence theLandau gauge gluon is unphysical, and it can be interpretedas a reflection of confinement. In [18,19], it was discussedhow

Dðk2Þ ¼ k2 þM2

k4 þ ðM2 þm2Þk2 þ �4(1.2)

can qualitatively and quantitatively describe the latticepredictions for the (infrared) Landau gauge gluon propa-gator, and this up to k� 1:5 GeV. The expression (1.2) canbe seen as a dynamically improved version of the so-calledGribov propagator

Dðk2Þ ¼ k2

k4 þ �4; (1.3)

which attracted a lot of attention as a prototype of aconfining gluon propagator and which is related to thedealing with the issue of gauge (Gribov) copies in theLandau gauge [33–35]. Given that (1.2) works out well,it might subsequently be used in explicit computations, forexample, when one would like to learn something on theglueball or meson spectrum.

We notice that the propagator (1.2) can be decomposedinto 2 Yukawa propagators, with either 2 real or 2 complexconjugate masses, depending on the relative size of thescales present. The case of 2 complex conjugate massesseems to be realized [19], and a (tree level) positivityviolation is then guaranteed. The unphysical degrees offreedom corresponding to those complex conjugate masseswere called i particles [36]. Also the so-called Stinglpropagator [37,38]

Dðk2Þ ¼ k2

k4 þm2k2 þ �4(1.4)

can exhibit a pair of complex conjugate poles, as a specialcase of (1.2).

If we are interested in a spectral function inspired ap-proach to e.g. bound states in theories with propagators ofthe type (1.2), we of course would need to know how tocompute with such a propagator. Research in this directionwas initiated with the work on a toy model [36], where thenecessary spectral densities were computed by clever ma-nipulations, which unfortunately do not have general ap-plicability, or, more realistically, for gauge theories in[39,40] by using the Cutkosky cut rules and assumingthat the results can be continued to Euclidean space andthe case of complex conjugate masses.

The main purpose of this paper is to show that the cutrules results, which we summarized in Sec. II, are actuallycorrect. We shall therefore bring to attention in Sec. III theStieltjes integral transformation, as well as its inverse, asstudied by Widder [41], which allows us to compute di-rectly in Euclidean space the spectral density, even in caseswhen there is no cut rules prescription. We shall work out

an introductory example in Sec. IV. In Sec. V, we commenton the use of subtracted spectral representations to avoiddivergences. We then present in Sec. VI in full detail themost interesting case of 2 i-particle propagators with com-plex conjugate masses, wherefore a completely analyticaltreatment can also be given. In our concluding remarks,Sec. VII, we point out that theWidder result to compute thespectral function could perhaps also be used in a numericalcontext when more complicated propagators than e.g. (1.2)are used, which are completely inadequate for fully ana-lytical computations.

II. USING THE CUTKOSKY CUT RULES

A. Preliminaries

We assume that O is a generic (composite) operatorcontaining 2 fields. We are then interested in the study ofthe following 1-loop expression:

hOðkÞOð�kÞi ! F ðk;m1; m2Þ

¼Z ddp

ð2�Þd1

ðk� pÞ2 �m21

1

p2 �m22

� fðp; k� pÞ; (2.1)

where fðp; k� pÞ is a function of the Lorentz invariantsconstructed from the momenta flowing in the loop. Forsimplicity, we may assume O to be a scalar function, thusa fortiori also fðp; k� pÞ. If O would carry Lorentzindices, say, O ¼ O�1...�n

, the eventual correlation func-

tion hO�1...�nðkÞO�1...�n

ð�kÞi ¼ F �1...�n�1...�nðk; m1; m2Þ

would be a tensor object. One can always decomposethis tensor into a suitably constructed basis (using thevector k� and trivial unit tensors), and by projecting on

the basic tensor structures, one can construct the spectraldensities associated to each structure. The latter problemcan then be reduced to that of the scalar case by contrac-tions. For example, assume that the two-point function is arank-2 tensor F��ðkÞ. Each rank-2 tensor can be decom-

posed into its transversal and longitudinal part, accordingto

F ��ðkÞ ¼ F TðkÞ�g�� �

k�k�

k2

�|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

P��ðkÞ

þF LðkÞ�k�k�

k2

�|fflfflffl{zfflfflffl}L��ðkÞ

; (2.2)

so that knowledge of F TðkÞ, respectively, F LðkÞ, followsfrom the (scalar) quantity P��ðkÞF��ðkÞ, respectively

L��ðkÞF ��ðkÞ. Using these scalar objects, one can then

construct the spectral densities corresponding to the trans-versal and longitudinal piece of the two-point function.The here described procedure can be generalized to morecomplicated tensorial structures.We thus wish to find the spectral representation (1.1) of

the integral (2.1), and we shall therefore use the Cutkosky

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cut rules. In order to do so, we must assume thatm21 andm

22

are real and that we are in Minkowski spacetime.By cutting the lines comprising this 1-loop diagram and,

according to the rules, performing the replacement [2,4]

1

ðp2 �m2i Þ

! 2��ðp0Þ�ðp2 �m2i Þ; (2.3)

the spectral function is proportional to the discontinuity,�ð�Þ / DiscF ð�Þ ¼ 2 ImF ð�Þ, and it can be determinedfrom

ImF ðk;m1; m2Þ ¼ 1

2

Z ddp

ð2�Þd ½ð2�Þ2�ððk� pÞ0Þ

� �ððk� pÞ2 �m21Þ�ðp0Þ�ðp2 �m2

2Þ� fðp; k� pÞ�: (2.4)

In order to evaluate (2.4) we will work in a frame wherek� ¼ ðk0; 0Þ ¼ ðE; 0Þ. We have then

ImF ðE2Þ ¼ 1

2

Z ddp

ð2�Þðd�2Þ ½�ðE� p0Þ�ððE� p0Þ2

�!2p;1Þ�ðp0Þ�ððp0Þ2 �!2

p;2Þfðp; k� pÞ�;(2.5)

where !p;i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~p2 þm2

i

q. In the evaluation of this integral

one must observe that the function fðp; k� pÞ comesfrom possible derivatives in the operator OðkÞ and thatit is a Lorentz scalar. It can thus only be a functionof E, m1, and m2, which comes from the scalars p2 ¼m2

2, ðk� pÞ2 ¼ m21, and p � ðk� pÞ ¼ 1

2 ðE2 �m21 �m2

2Þ.After the evaluation we can write the Lorentz invariantspectral function �ð�Þ by going to an arbitrary frame, whilereplacing E2 ! �.

We are ultimately interested, however, in the case wherethe masses involved are complex and the momenta are inEuclidean space. We then consider expressions like (2.5) asfunctions of real masses and Minkowski momenta and willanalytically continue1 it to complex mass values inEuclidean momentum space. We want to draw attentionhere that, if wewould immediately assumem2

1 andm22 to be

complex, we cannot apply the cut rules. We cannot put an iparticle physically on-shell, or mathematically spoken,(2.3) does not make much sense form2

i 2 Cwhile p2 2 R.

B. Naive application of the cut rules

We will now illustrate the procedure just described bycomputing spectral functions in different dimensionalitiesfor the two-point function (2.1) with fðp; k� pÞ ¼ 1. Thiscorresponds to the toymodel introduced in [36].We are thusinterested in finding a spectral representation of the form

Fdðk2Þ ¼Z 1

�0

d��dð�Þ�þ k2

(2.6)

for the following Euclidean two-point function:

Fdðk2Þ ¼Z ddp

ð2�Þd1

p2 þm21

1

ðp� kÞ2 þm22

; (2.7)

with conjugate masses parametrized as

m21 ¼ aþ ib; m2

2 ¼ a� ib: (2.8)

We assume that2 a � 0, and without loss of generality wemay take b � 0.We shall therefore first use the Minkowski formula (2.5)

and integrate over p0 in (2.5), obtaining

ImF ðE2Þ ¼ 1

2

Z dðd�1Þpð2�Þðd�2Þ

1

2!p;2

½�ðE�!p;2Þ

� �ððE�!p;2Þ2 �!2p;1Þ�: (2.9)

The integrand depends only on ~p2 so that we can write

ImF ðE2Þ ¼ 1

2

1

ð2�Þðd�2Þ2�d�1=2

�ðd�12 Þ

Z 1

0dj ~pjj ~pjðd�2Þ

� 1

2!p;1

1

2!p;2

�ðE�!p;1 �!p;2Þ; (2.10)

where we also took into account the theta function.The remaining delta function can be cast in a moreconvenient form using the property �ðgð ~p2ÞÞ ¼ 1

jg0ð ~p20Þj�

�ð ~p2 � ~p20Þ ¼ 1

2j ~p0jjg0ð ~p20Þj�ðj ~pj � j ~p0jÞ, where ~p2

0 is such

that gð ~p20Þ ¼ 0. In our case, ~p2

0 satisfies

gð ~p20Þ � E�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~p20 þm2

1

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~p20 þm2

2

q¼ 0: (2.11)

The integral (2.10) can now be readily evaluated:

ImF ðE2Þ ¼ 1

ð2�Þðd�2Þ�d�1=2

�ðd�12 Þ

j ~p0jd�3

4E; (2.12)

where

j ~p0j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðE2 �m2

1 �m22Þ2 � 4m2

1m22

q2E

: (2.13)

Having done this, we can now pass to the Euclidean casewith the masses given by (2.8), by evaluating this expres-sion (2.13),

j ~p0j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE4 � 4b2 � 4aE2

p

2E: (2.14)

1A certain care is needed when speaking about ‘‘analyticcontinuation,’’ as we are considering a function depending ontwo variables, viz. m2

1 and m22.

2This in order to avoid tachyon instabilities. For b ¼ 0, this isimmediately clear. In the context of the model of [18], a itselfalso corresponds to the mass of another degree of freedom, andas such a is supposed to be positive.

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The expression (2.12) can then be worked out further forthe various dimensions. For d ¼ 2 we obtain

ImF d¼2ðE2Þ ¼ 1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE4 � 4b2 � 4aE2

p : (2.15)

We eventually conclude that

�d¼2ð�Þ ¼ 1

2�

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 4b2 � 4a�

p ; (2.16)

using E2 ! � and the equivalence �ð�Þ ¼ 1� ImF ð�Þ.

Analogously, for d ¼ 3 and d ¼ 4, we find

ImF d¼3ðE2Þ ¼ 1

8E! �d¼3ð�Þ ¼ 1

8�

1ffiffiffi�

p ; (2.17)

ImF d¼4ðE2Þ ¼ 1

16�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4b2

E4� 4a

E2

s! �d¼4ð�Þ

¼ 1

ð4�Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4b2

�2� 4a

�

s: (2.18)

The threshold �0 is in all cases given by

�0 ¼ ðm1 þm2Þ2 ¼ 2ðaþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

pÞ: (2.19)

III. SURVEY OF THE WIDDER STIELTJESINVERSION OPERATOR

In this section, we shall first refresh the concept ofthe Stieltjes integral transformation and then discusshow the inverse transformation can be found.

Let us assume a function FðxÞ, defined for x > 0, andimplicitly define another function �ðtÞ by means of

FðxÞ ¼Z þ1

0dt

�ðtÞtþ x

; (3.1)

where we assume that the latter integral exists. This op-eration defines the Stieltjes integral transformation [41,42].Upon comparing (1.1) and (3.1), it is clear that a Stieltjesrepresentation (3.1) of the function FðxÞ is nothing elsethan the Kallen-Lehmann representation if FðxÞwould be atwo-point function in Euclidean space. If �ðtÞ ¼ 0 fort < A, A sets the threshold �0.

The spectral density �ðtÞ can be reconstructed from[41,42]

�ðtÞ ¼ limn!þ1ð�1Þnþ1 1

ðn!Þ2 @nt ½t2nþ1@nþ1

t FðtÞ�: (3.2)

Let us first prove this statement along the lines of [41], asthe tools of the proof shall turn out to be useful for our lateranalysis too. We start with

ð�1Þnþ1 1

ðn!Þ2 @nt ½t2nþ1@nþ1

t FðtÞ�

¼ ð�1Þnþ1 1

ðn!Þ2 @nt

�t2nþ1@nþ1

t

Z þ1

0du

�ðuÞtþ u

�

¼ nþ 1

n!@nt

�t2nþ1

Z þ1

0du

�ðuÞðtþ uÞnþ2

�

¼ nþ 1

n!

Z þ1

0du�ðuÞun@nt

�t2nþ1

unðtþ uÞnþ2

�: (3.3)

It seems to be more involved to compute the latter deriva-tive, but we can use a nice trick to do so [41]. Defining

gðt; uÞ ¼ t2nþ1

unðtþ uÞnþ2; (3.4)

we notice that this is a homogenous function of order �1,as for any ‘ > 0,

gð‘t; ‘uÞ ¼ ‘�1gðt; uÞ: (3.5)

Deriving with respect to ‘ and setting ‘ ¼ 1 leads to theEuler characterization of homogenous functions.Specifically,

t@tgþ u@ug ¼ �g; (3.6)

or for t > 0,

@tg ¼ �@u

�u

tg

�: (3.7)

If gðt; uÞ is homogenous of order �1, so is ut gðt; uÞ, and

hence we can iteratively employ (3.7) to compute

@nt gðt; uÞ ¼ ð�1Þn@nu�un

tngðt; uÞ

�

¼ ð�1Þntnþ1@nu1

ðtþ uÞnþ2

¼ tnþ1 ð2nþ 1Þ!ðnþ 1Þ!

1

ðtþ uÞ2nþ2: (3.8)

We are thus lead to

ð3:3Þ ¼ ð2nþ 1Þ!ðn!Þ2

Z þ1

0du

�ðuÞu

�ut

ðuþ tÞ2�nþ1

¼ ð2nþ 1Þ!ðn!Þ2

Z þ1

0du

�ðuÞu

eðnþ1Þhðu;tÞ; (3.9)

with

hðu; tÞ ¼ lnut

ðuþ tÞ2 : (3.10)

In the limit n ! 1, the latter integral is ideally suited for asteepest descent evaluation, since

@uhðu;tÞju¼t¼0; @2uhðu;tÞju¼t¼� 1

2t2<0: (3.11)

Doing so, we find

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limn!þ1

ð2nþ 1Þ!ðn!Þ2

Z þ1

0du

�ðuÞu

eðnþ1Þhðu;tÞ

¼ limn!þ1

ð2nþ 1Þ!ðn!Þ2

� �ðtÞt

eðnþ1Þhðt;tÞ Z þ1

0due�ðnþ1Þððu�tÞ2=4t2Þ; (3.12)

with

hðt; tÞ ¼ �2 ln2; (3.13)

and, for n ! þ1,Z þ1

0due�ðnþ1Þððu�tÞ2=4t2Þ !

Z þ1

�1due�ðnþ1Þððu�tÞ2=4t2Þ

¼ 2

ffiffiffiffi�

pffiffiffiffiffiffiffiffiffiffiffiffi1þ n

p t: (3.14)

Recalling Stirling’s formula

n! ! ffiffiffiffiffiffiffiffiffi2�n

p �n

e

�n; for n ! þ1; (3.15)

we can write that

ð2nþ1Þ!ðn!Þ2 ! 1

2ffiffiffiffi�

p 4nþ1ffiffiffiffiffiffiffiffiffiffiffinþ1

p; for n!þ1: (3.16)

We then find

limn!þ1

ð2nþ 1Þ!ðn!Þ2

�ðtÞt

eðnþ1Þhðt;tÞ Z 1

0due�ðnþ1Þððu�tÞ2=4t2Þ

¼ �ðtÞ; (3.17)

hereby proving (3.2).Once having determined �ðtÞ via (3.2), one can then

define

FðzÞ ¼Z þ1

0dt

�ðtÞtþ z

(3.18)

for z 2 C, which is analytic, with the exception of a branchcut on the negative real axis, z � 0 [42]. Obviously, therewill be no branch cut for z 2 ½��2;��1� if �ðtÞ ¼ 0 fort 2 ½�1; �2�, with �1 > �2 � 0. Using Cauchy’s formula,one can then see that it actually holds that

�ðtÞ ¼ 1

2�ilim!0þ

½Fð�t� iÞ � Fð�tþ iÞ�; (3.19)

which corresponds to the discontinuity of FðzÞ along thenegative real axis. En route, this explains the dispersionrelation (1.1).

Apparently, (3.19) would provide us a with a mucheasier way to compute the spectral density �ðtÞ, but thedifficulty is that in most cases, we do not know how toevaluate FðzÞ from its original (integral) definition forz =2 Rþ

0 . In contrast with this, (3.2) only needs FðzÞ forz 2 Rþ

0 . This is exactly what we need, since we can always

evaluate the Euclidean momentum integrals defining a

two-point function Fðk2Þ, in which case we can assumek2 2 Rþ

0 . Afterwards, FðzÞ is defined by means of (3.18)

for all z 2 C.In principle, (3.18) also makes sense when �ðtÞ is a

distribution. A basic example is �ðtÞ ¼ �ðt�m2Þ, whichleads to the familiar Yukawa propagator Fðk2Þ ¼ 1

k2þm2 ,

which evidently has no branch cut.

IV. A FIRST APPLICATION: TWO (POSITIVE)REAL MASSES

As a warming up exercise, we shall treat here a well-known textbook example and compute the spectral densityin the case of 2 real masses. This was already treated in [3],albeit in Minkowski space. We shall work in Euclideanspace, and we start from (2.7). We introduce a Feynmanparameter x, yielding

Fdðk2Þ ¼Z 1

0dx

Z ddp

ð2�Þd

� 1

½xððk� pÞ2 þm21Þ þ ð1� xÞðp2 þm2

2Þ�2:

(4.1)

The substitution q ¼ p� xk gives

Fdðk2Þ ¼Z 1

0dx

Z ddq

ð2�Þd1

ðq2 þ�2Þ2 ; (4.2)

with

�2 ¼ xk2 þ xm21 þm2

2 � xm22 � x2k2: (4.3)

Usage of the standard Euclidean space formula

Z ddq

ð2�Þd1

ðq2 þ �2Þn ¼1

ð4�Þd=2 ð�2Þd=2�n �ðn� d=2Þ

�ðnÞ(4.4)

leads to

Fdðk2Þ ¼ 1

ð4�Þd=2 ð�2Þd=2�2�ð2� d=2Þ: (4.5)

For the rest of this section, we shall mainly concentrateourselves on the d ¼ 2 case. We are thus interested in

Fd¼2ðk2Þ ¼ 1

4�

Z 1

0

dx

xð1� xÞk2 þ xðm21 �m2

2Þ þm22

:

(4.6)

Setting t ¼ k2 and dropping the irrelevant prefactor, wemay focus on

FðtÞ ¼Z 1

0

dx

xð1� xÞk2 þ xðm21 �m2

2Þ þm22

: (4.7)

Subsequently,

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@nþ1t FðtÞ ¼ @nþ1

t

Z 1

0

dx

xð1� xÞ1

tþ

¼ ð�1Þnþ1ðnþ 1Þ!Z 1

0

dx

xð1� xÞ1

ðtþ Þnþ2;

(4.8)

where we temporarily set � ðxÞ ¼ xðm21�m2

2Þþm2

2

xð1�xÞ . Doing

so, we have

�ðtÞ ¼Z 1

0

dx

xð1� xÞ limn!þ1

nþ 1

n!@nt

�t2nþ1

ðtþ Þnþ2

�: (4.9)

We already computed the derivative appearing in the right-hand side of (4.9), and reusing the result (3.8) yields

�ðtÞ¼ limn!þ1

ð2nþ1Þ!ðn!Þ2

Z 1

0

dx

xð1�xÞntnþ1

ðtþÞ2nþ2

¼ limn!þ1

ð2nþ1Þ!ðn!Þ2

Z 1

0

dx

xð1�xÞ1

eðnþ1Þhð;tÞ: (4.10)

We notice the great power of this formulation, as thespectral density can now be obtained from a steepestdescent evaluation of the Feynman parameter integral(4.10). It is useful to perform the substitution x ¼ 1

1þy to

rewrite (4.10) as

�ðtÞ ¼ limn!þ1

ð2nþ 1Þ!ðn!Þ2

�Z þ1

0

dy

ð1þ yÞðm21 þ ym2

2Þeðnþ1Þhð;tÞ; (4.11)

with

� ðyÞ ¼ 1þ y

yðm2

1 þ ym22Þ: (4.12)

We are now ready to search for the maxima of hðy; tÞ ¼hððyÞ; tÞ for y 2 ½0;1�. Solving @yhðy; tÞ ¼ 0 gives

y1 ¼ m1

m2

;

y2 ¼t�m2

1 �m22 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4m2

1m22 þ ð�tþm2

1 þm22Þ2

q2m2

2

;

y3 ¼ �m1

m2

;

y4 ¼t�m2

1 �m22 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4m2

1m22 þ ð�tþm2

1 þm22Þ2

q2m2

2

:

(4.13)

We can discriminate between a few cases.(1) 0< t < ðm1 �m2Þ2.—In this case, the root appear-

ing in (4.13) exists. However, only y1 > 0. We have

@2yhjy¼y1 ¼2m3

2ðt� ðm1 þm2Þ2Þm1ðm1 þm2Þ2ðtþ ðm1 þm2Þ2Þ

� ��2 < 0:

(4.14)

We can then look at the behavior of the integral forn ! þ1,Z þ1

0

dy

ð1þ yÞðm21 þ ym2

2Þeðnþ1Þhð;tÞ

/ 1ffiffiffiffiffiffiffiffiffiffiffiffinþ 1

p eðnþ1Þhðy1;tÞ: (4.15)

Using

hðy1; tÞ ¼ lntðm1 þm2Þ2

ðtþ ðm1 þm2Þ2Þ2; (4.16)

we findZ þ1

0

dy

ð1þ yÞðm21 þ ym2

2Þeðnþ1Þhð;tÞ

/ 1ffiffiffiffiffiffiffiffiffiffiffiffinþ 1

p�

tðm1 þm2Þ2ðtþ ðm1 þm2Þ2Þ2

�nþ1

: (4.17)

Using (3.16), we conclude that �ðtÞ ¼ 0 as 0 �tðm1þm2Þ2

ðtþðm1þm2Þ2Þ2 <14 .

(2) ðm1 �m2Þ2 < t < ðm1 þm2Þ2.—In this interval,the roots appearing in (4.13) are not real, so againonly y1 is relevant. The rest of the reasoning remainsvalid, so we conclude again that �ðtÞ ¼ 0.

(3) t > ðm1 þm2Þ2.—In this case, the roots are againreal-valued. Looking at (4.18), we notice that y1 nowcorresponds to a minimum, so we can disregard it.Only y2 and y4 are relevant, as both are positive andcorrespond to local maxima. We shall not writedown some intermediate results, as they are quitelengthy. We set

@2yhjy¼y2 ¼ ��22 < 0 ¼ @2yhjy¼y4 ¼ ��2

4 < 0:

(4.18)

It then suffices to notice that

hðy2; tÞ ¼ hðy3; tÞ ¼ �2 ln2; (4.19)

which is exactly the value we need to find a non-vanishing finite contribution to the integral. In par-ticular,

�ðtÞ ¼ 1

ð1þ y2Þðm21 þ y2m

22Þ

ffiffiffi2

p2�2

þ 1

ð1þ y4Þðm21 þ y4m

22Þ

ffiffiffi2

p2�4

; (4.20)

which simplifies to

DAVID DUDAL AND MARCELO S. GUIMARAES PHYSICAL REVIEW D 83, 045013 (2011)

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�ðtÞ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4m2

1m22 þ ðt�m2

1 �m22Þ2

q (4.21)

after a bit of algebra.At the end of the day, we find that

Fd¼2ðk2Þ ¼Z þ1

ðm1þm2Þ2dt

�ðtÞtþ k2

; with

�ðtÞ ¼ 1

2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4m2

1m22 þ ðt�m2

1 �m22Þ2

q : (4.22)

It is easily verified that �ðtÞ � 0 for t � ðm1 þm2Þ2.These results are in full agreement with those obtainedfrom the Cutkosky rules (see also [3]) given that we trans-form from Minkowski to Euclidean space.

V. A COMMENTABOUT DIVERGENTSPECTRAL INTEGRALS

Setting d ¼ 4 in (4.5), we clearly stumble upon a diver-gence / �ð0Þ. In order to deal with finite quantities, oneusually turns to a suitably subtracted spectral representa-tion. In particular, if we formally set

GðxÞ ¼Z 1

0dt

�ðtÞtþ x

ð¼ 1Þ (5.1)

for a divergent3 quantity GðxÞ, it is clear that a suitablenumber of derivatives with respect to x will render us witha finite result

HðxÞ ¼ @N

ð@xÞN GðxÞ ¼ ð�1ÞNZ 1

0dt

�ðtÞðtþ xÞNþ1

<1;

(5.2)

leading to a finite subtracted spectral representation afterNintegrations from 0 to x,

GsubðxÞ ¼ GðxÞ � � � � � xn�1

ðN � 1Þ!@N�1G

@xN�1ð0Þ

¼ ð�1ÞNxNZ 1

0dt

�ðtÞtNðtþ xÞ<1: (5.3)

In order to know the spectral representation correspondingto GðxÞ, it looks preferential to study the finite functionHðxÞ, (5.2), and bring it into a spectral representation of theform

HðxÞ ¼ ð�1ÞNZ 1

0dt

�ðtÞðtþ xÞNþ1

; (5.4)

from which we can also read off the desired spectraldensity �ðtÞ of the original function GðxÞ. Using the tech-niques of Sec. II, it is not difficult to show that4

�ðtÞ ¼ limn!þ1ð�1Þnþ1�N N!

ðn!Þ2 @nt ½t2nþ1@nþ1�N

t HðtÞ�;(5.5)

a formula which allows us to compute the spectral density�ðtÞ from knowledge of the quantityHðxÞ, and this withoutencountering any infinities at any time.In practice, glancing again at (4.5), we would have

@Fd¼4ðk2Þ@k2

¼ � 1

16�2

Z 1

0dx

1

k2 þ xðm21�m2

2Þþm2

2

xð1�xÞ; (5.6)

and upon comparing with (4.6), we notice that these arealmost identical. Only the ‘‘prefactor function’’ of the 1

tþ

will be different. In particular, this means that the saddlepoint equation, etc., for d ¼ 4 remains the same as ford ¼ 2, just as almost all the rest of the derivation. Onlyduring the last step in computing the spectral density shallwe notice a difference, as only there the prefactor functionplays a role.The case d ¼ 3 would need to be worked out from the

beginning, but similar tricks as for d ¼ 2 or d ¼ 4 can beapplied. This can be easily appreciated as a systematics in

the derivatives of 1=ffiffiffiffiffiffiffiffiffiffiffiffitþ

pexists, the latter being the

basic form that will appear in the d ¼ 3 case.

VI. A SECOND APPLICATION: TWO COMPLEXCONJUGATE MASSES

We now come to the most interesting application of thispaper. We reconsider the integral (2.7), rewritten in theform (4.6), but now we immediately assume that the occur-ring mass scales are complex conjugate and given by (2.8).It can be checked that the ‘‘trick-inspired’’ approaches of[36] to construct the spectral density no longer work out inthe case that b � 0. We wish to confirm that the results(2.16) and (2.19) obtained via a blind trust in the Cutkoskyapproach are actually correct.We start our analysis from the first line of (4.10). We

have

�ðtÞ ¼ limn!þ1

ð2nþ 1Þ!ðn!Þ2

Z 1

0

dx

xð1� xÞntnþ1

ðtþ Þ2nþ2; (6.1)

now with

� ðxÞ ¼ xðm21 �m2

2Þ þm22

xð1� xÞ¼ a

xð1� xÞ þ ib2x� 1

xð1� xÞ : (6.2)

In principle, one might try to do a steepest descent evalu-ation of the integral (6.1), keeping in mind that this timeone would need to deform the integration contour to followa line of steepest descent through the equivalent ofthe saddle points (4.13) in the complex x plane. This looksas a very complicated exercise, instead we shall follow a

3We assume here that the divergency is due to the large tbehavior of �ðtÞ, something which is usually the case.

4We may always take n � N.

COMPUTATION OF THE SPECTRAL DENSITY OF TWO- . . . PHYSICAL REVIEW D 83, 045013 (2011)

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somewhat different reasoning. Let us first notice that (6.1)defines a real function, as

Z 1

0

dx

xð1� xÞntnþ1

ðtþ Þ2nþ2

¼Z 1=2

0

dx

xð1� xÞ1

�t

ðtþ Þ2�nþ1

þZ 1

1=2

dx

xð1� xÞ1

�t

ðtþ Þ2�nþ1

¼Z 1=2

0

dx

xð1� xÞ�1

�t

ðtþ Þ2�nþ1

�

þZ 1=2

0

dx

xð1� xÞ�1

�t

ðtþ Þ2�nþ1

�;

via the substitution x ! 1� x in the 2nd integral. We canthus say

�ðtÞ ¼ 2Re½RðtÞ�; (6.3)

and it suffices to study

RðtÞ ¼ limn!þ1

ð2nþ 1Þ!ðn!Þ2

Z 1=2

0

dx

xð1� xÞ1

�t

ðtþ Þ2�nþ1

:

(6.4)

It would be beneficial if we could use as a (complex)integration variable, whereby we integrate along the con-tour5 �, which has the parameter representation

�: x 2 ½0; 1=2� ! ðxÞ ¼ a

xð1� xÞ þ ib2x� 1

xð1� xÞ :(6.5)

This � will start from a point at complex infinity in theright lower half plane and will end on the real axis at ¼ 4a. Inverting gives

x ¼ �2ibþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4b2 � 4aþ 2

p

2: (6.6)

A careful examination learns that the minus sign is theappropriate choice, given thatðxÞ lives in the lower halfplane for x 2 ½0; 1=2�. We can subsequently reexpress(6.4) as

RðtÞ ¼ limn!þ1

ð2nþ 1Þ!ðn!Þ2

Z�

�dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4b2 þ ð�4aþ Þp� 1

�t

ðtþ Þ2�nþ1

(6.7)

after a little algebra. For the further analysis, it is easier toswitch the orientation of the contour �, so that we can write

RðtÞ ¼ limn!þ1

ð2nþ 1Þ!ðn!Þ2

Z�

dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4b2 þ ð�4aþ Þp� 1

eðnþ1Þ lnhð;tÞ; (6.8)

with � as shown in Fig. 1 and hð; tÞ given by (3.10).The presence of the square root in (6.8) complicates the

situation a bit. Setting ¼ x þ iy learns there is a

‘‘crucifix cut’’ given by

x ¼ 2a; y 2 R;

x 2 ½2ða�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

pÞ; 2ðaþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

pÞ�; y ¼ 0:

(6.9)

Our goal is now to compute (6.8) by means of a steepestdescent approach. We can use previously gained knowl-edge, from which we learn that hð; tÞ has a saddle point at ¼ t. In the vicinity of the saddle point, we may write

lnhð; tÞ ¼ �2 ln2� ð� tÞ24t2

þ � � � : (6.10)

We can now consider a few cases.

(1) t > 2ðaþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

pÞ.—Since everything is analytic

in the considered region of the complex plane,6 wecan deform our contour � into �0, which shares itsbegin and end point with �, such that �0 passesthrough the saddle point. We have to choose theorientation of the contour �0 in the vicinity of thesaddle point in such a way that the imaginary part oflnh remains constant. From (6.10), it is easily seenthat this is the case when we take real for a while

FIG. 1. Branch cuts of the integrand of Eq. (6.8), the integra-tion contour �, and its deformation �0.

5We do not necessarily refer to a closed contour here.

6The presence of the ln is of no concern for the analyticity,since n 2 N.

DAVID DUDAL AND MARCELO S. GUIMARAES PHYSICAL REVIEW D 83, 045013 (2011)

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until we have passed t, and then we can let �0 bendover to let it flow to the correct end point at infinity.7

Doing so, we find

limn!1

Z�

dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4b2 þ ð�4aþ Þp 1

eðnþ1Þ lnhð;tÞ

¼ limn!1e

�2ðnþ1Þ ln2 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4b2 þ tð�4aþ tÞp 1

t

�Z þ1

�1dxe

�ðnþ1Þ ðx � tÞ24t

¼ limn!1

1

4nþ1

2ffiffiffiffi�

pffiffiffiffiffiffiffiffiffiffiffiffi1þ n

p 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4b2 þ tð�4aþ tÞp ;

(6.11)

where we made use of (3.14). Combining this resultwith (3.16) gives

RðtÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4b2 þ tð�4aþ tÞp : (6.12)

Hence,

�ðtÞ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4b2 þ tð�4aþ tÞp : (6.13)

(2) 4a < t < 2ðaþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

pÞ.—In this region, we can

still deform the contour � into an appropriate �0 in acompletely similar way, as we do not have to crossany cut in order to do so. Since we are below the cut,we find after a completely analogous reasoning as inthe first case

RðtÞ ¼ iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4b2 þ tð4a� tÞp ; (6.14)

so that now

�ðtÞ ¼ 0: (6.15)

(3) 2a < t < 4a.—At first sight, we can repeat theanalysis of subcase 2. This reasoning is, however,flawed. A basic ingredient of the steepest descentapproach is that Reðlnhð; tÞÞ reaches its maximumalong the (deformed) contour at the value of thesaddle point, in this case ¼ t. If we have to firststretch our contour � into the left direction to pickup the saddle point, we are violating this assump-tion, as it can be checked. The gray region in Fig. 2displays the region in the complex plane wecannot cross with our contour.The boundaries of these curves are defined by

y ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7t2 � 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit3ð3t� 2xÞ

q� 2tx � 2

x

r;

y ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7t2 � 2tx � 2

x þ 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3t4 � 2t3x

qr:

(6.16)

If t > 4a, the begin point of � lies within the whiteblob, and we can (only) follow the x axis, pick upthe saddle point at x ¼ t, leave this region, and goto infinity. If, however, we start outside this region,there is no way that we can go through the saddlepoint and go back to infinity without self-crossingthe contour. Said otherwise, the steepest descentapproach is inapplicable in this case.

(4) 0 � t < 2a.—In this region, we seem to be in evenmore serious trouble, as we can no longer evendeform our contour in a useful way, due to thevertical branch cut at ¼ 2a.

(5) 0 � t < 4a.—Let us now present a combined studyof the two previous cases, based on a differentapproach. Using the triangle inequality for integrals,we have

jRðtÞj¼ limn!þ1

ð2nþ1Þ!ðn!Þ2

��������Z 1=2

0

dx

xð1�xÞntnþ1

ðtþÞ2nþ2

��������� lim

n!þ1ð2nþ1Þ!ðn!Þ2

Z 1=2

0

dx

xð1�xÞ�������� ntnþ1

ðtþÞ2nþ2

��������¼ lim

n!þ1ð2nþ1Þ!ðn!Þ2

Z 1=2

0

dx

xð1�xÞ1

jj

��

tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2þb2ð1�2xÞ2p

xð1�xÞða�txð1�xÞÞ2þb2ð1�2xÞ2

�nþ1

:

(6.17)

We consequently observe that the (positive) func-tion

FIG. 2. Forbidden (gray) and allowed (white) regions in thecomplex plane that the contour � (or �0) can and cannot crossfor the steepest descent evaluation.

7As the function vanishes at infinity, we can also move the endpoint at infinity, if desired.

COMPUTATION OF THE SPECTRAL DENSITY OF TWO- . . . PHYSICAL REVIEW D 83, 045013 (2011)

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jðx;tÞ¼ tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2þb2ð1�2xÞ2p

xð1�xÞða� txð1�xÞÞ2þb2ð1�2xÞ2 (6.18)

has maxima at the solutions of

@j

@x¼ @j

@t¼ 0; (6.19)

namely,

ðx; tÞ ¼ ð0; 0Þ; ðx; tÞ ¼ ð1; 0Þ;ðx; tÞ ¼ ð1=2; 4aÞ; (6.20)

with

jð0; 0Þ ¼ 0; jð1; 0Þ ¼ 0; jð1=2; 4aÞ ¼ 14:

(6.21)

On the interval ðx; tÞ 2 ½0; 1=2� � ½0; 2a�, this im-plies that, for a certain ,

0 � jðx; tÞ � < 14; (6.22)

and using this we can majorate (6.17) further as

jRðtÞj� limn!þ1

ð2nþ1Þ!ðn!Þ2 nþ1

Z 1=2

0

dx

xð1�xÞ1

jj|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}number independentof n

:

(6.23)

Once more using (3.16), we then simply find

jRðtÞj � 0; (6.24)

as < 14 .A fortiori,RðtÞ ¼ 0, and thus also�ðtÞ ¼ 0.

Combining all information gathered so far, we haveshown that

Fd¼2ðk2Þ ¼Z þ1

2ðaþffiffiffiffiffiffiffiffiffiffia2þb2

pÞdt

�ðtÞtþ k2

; with

�ðtÞ ¼ 1

2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4b2 þ tð�4aþ tÞp ;

(6.25)

whereby we can observe that �ðtÞ � 0 for t � 2ðaþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

pÞ.

As a check on the final result (6.25), we notice that itsb ! 0 limit coincides with the m2

1 ! m22 ! a limit of the

earlier obtained result (4.22).For the general case, we could of course explicitly

compute the spectral integral of (6.25), as well as theoriginal Feynman parameter integral (4.6), and verify ifboth results are the same. We find it, however, moreinstructive for the reader to display in Fig. 3 both resultsfor the explicit example m2

1 ¼ 1þ 2i, m22 ¼ 1� 2i, mak-

ing their equivalence clearly visible. Similar conclusionscan be reached for d ¼ 3 or d ¼ 4. In all cases, we thusfind perfect agreement between the results obtained byapplication of the Cutkosky cut rules and those obtainedby using the Widder Stieltjes formalism.

VII. CONCLUDING REMARKS

We have demonstrated the usefulness of the Stieltjesintegral transform to compute the Kallen-Lehmann spec-tral density of (Euclidean) two-point functions, even incases where the validity of the usually employedCutkosky rules is not clear. The main recipes are given inSecs. IV and V, in particular, in expression (3.2), whichgives the spectral density �ðtÞ in terms of the two-pointfunction Fðk2Þ, when the latter is known for Euclideanmomenta k2 � 0.We have paid particular attention to the case of Gribov-

like propagators, which entails the presence of propagatorswith 2 complex conjugate masses. It turns out that, at leastat one loop, this case can be worked out analytically to the

1 2 3 4 5k2

0.45

0.50

0.55

F1 k2

(a)

1 2 3 4 5k2

0.45

0.50

0.55

F2 k2

(b)

FIG. 3 (color online). Plots of F1ðk2Þ ¼R10

dxxð1�xÞtþ2ibxþa�ib and F2ðk2Þ ¼

Rþ12ðaþ

ffiffiffiffiffiffiffiffiffiffia2þb2

pÞ dt

�ðtÞtþk2

for a ¼ 1 and b ¼ 2.

DAVID DUDAL AND MARCELO S. GUIMARAES PHYSICAL REVIEW D 83, 045013 (2011)

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end using a steepest descent approach. Our main conclu-sion is that the results, obtained by the cut rules for realmasses and in Minkowski space, may be continued toEuclidean space, while also the two real masses can for-mally be replaced by two complex conjugate values. Thiskind of result already found use in recent works like[39,40] and has now been proven to be correct.

However, the inversion formula (3.2) is also valid incases where the cut rules are unavailable. We recall thatmany works have been devoted to the study of the quantumequations of motion (Schwinger-Dyson formalism), givingalso estimates for the gluon propagator, among other quan-tities [9,10,15–17,21]. In the deep infrared and ultraviolet,an analytical approach is possible. In the intermediatemomentum regime, however, numerical solutions are inorder due to the complexity of the system. The numericaloutcome can then fitted by well-guessed functional forms.Let us mention two major examples:

Dðk2Þ ¼ 1

k2 þm2ðk2Þ ;

m2ðk2Þ ¼ m40

k2 þm20

�ln

k2þfðk2;m20Þ

�2

lnfð0;m2

0Þ

�2

��3=5;

fðk2; m20Þ ¼ �1m

20 þ �2

m40

q2 þm20

;

�1 ¼ � 1

2; �2 ¼ 5

2;

m0 ¼ 612 MeV; � ¼ 645 MeV; (7.1)

which represents the so-called ‘‘massive’’ [15,43] (alsoknown as ‘‘decoupling’’ [17]) solution, or

Dðk2Þ ¼ ZI;IIðk2Þ�

ð0Þ1þ k2=�2

þ 4�

�0

k2

k2 þ�2

��

1

lnk2=�2� �2

k2 ��2

��13=22

;

�0 ¼ 11N

3; ð0Þ ¼ 8:915

N; � ¼ 710 MeV;

(7.2)

with

ZIðk2Þ¼wI

ðk2Þ2�ðk2Þ2�þð�2Þ2� ; or

ZIIðk2Þ¼wII

�k2

k2þ�2

�2�; ��0:595; (7.3)

for the so-called ‘‘scaling’’ solution [44].It is evidently out of the question to apply the standard

cut rules with the foregoing propagators, since they are noteven of the form 1

p2þm2 , so the replacement rule (2.3) loses

its meaning. The theory of the (Stieltjes) inversion could,however, be applied, albeit perhaps rather in a numericalfashion, employing stable approximations for the infinitenumbers of derivatives appearing in formula (3.2). Let usalso mention here that in many cases, the concrete appli-cation of the Cutkosky rules become intractable beyondone loop, due to the highly complicated phase spaceintegrals.In the particular case of bound states, like a glueball or

meson, which can crucially depend on a viable input for thegluon propagator, one might imagine to use propagators ofthe type (7.1) or (7.2) to compute, in one approximation oranother, suitable two-point functions, to obtain conse-quently an estimate for the spectral density, and theneventually use the latter as input for whatever methodone likes to employ to find estimates for e.g. bound statemasses. One might think about employing Laplacian sumrules as in [6] or techniques based on Pade approximants asin [40,45], to name only a few approaches to the boundstate problem whereby the spectral density enters.We end by mentioning that (7.1) or (7.2) are quite differ-

ent in the deep infrared as (7.1) tends to a strictly positiveconstant for k2 ! 0, while (7.2) vanishes in the same limit.The former scenario is recovered on the lattice [12–14].One could nevertheless wonder if both solutions couldreproduce more or less the same physics, as preliminarilyinvestigated in [46] in a specific example and approxima-tion. We hope to come back to this issue in the future, usingthe tools developed in this paper. The content of Sec. Vshould be of particular interest, as also the propagators(7.1) or (7.2) will lead to divergent spectral integrals.

ACKNOWLEDGMENTS

We thank S. P. Sorella, N. Temme, and N. Vandersickelfor useful discussions. D.D. is supported by the Research-Foundation Flanders (FWO Vlaanderen), while M. S. G.acknowledges financial support by FAPERJ, Fundacao deAmparo a Pesquisa do Estado do Rio de Janeiro. D.D.acknowledges the hospitality at the UERJ where this workwas initiated and finished.

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