Inverse amplitude method and Adler zeros

13 Pages·2008·0.3 MB·English

PHYSICALREVIEW D 77, 056006 (2008)
Inverse amplitude method andAdler zeros
A.Go´mez Nicola, J.R. Pela´ez, and G.R´ıos
Departamento de F´ısica Teo´rica II, Universidad Complutense, 28040Madrid, Spain
(Received 17 December 2007; published 11 March 2008)
Theinverseamplitudemethodisapowerfulunitarizationtechniquetoenlargetheenergyapplicability
region of effective Lagrangians. It has been widely used to describe resonances in hadronic physics,
combinedwithchiralperturbationtheory,aswellasinthestronglyinteractingsymmetrybreakingsector.
In this work we show how it can be slightly modified to also account for the subthreshold region,
incorporatingcorrectlytheAdlerzerosrequiredbychiralsymmetryandeliminatingspuriouspoles.These
improvements produce negligible effects on the physical region.
DOI: 10.1103/PhysRevD.77.056006 PACSnumbers: 11.55.Fv, 12.39.Fe, 13.75.Lb, 13.85.Dz
I. INTRODUCTION pling constants g and g0. The so-called equivalence theo-
rem(ET)[3]statesthatathighenergies(inR gauges,and
Effective field theories provide a systematic and model (cid:2)
to leading order in momenta over M and g and g0)
independent approach to systems whose symmetries and V
amplitudes involving longitudinal gauge bosons can be
low-energy degrees of freedom are known but whose de-
calculatedasiftheywereGB,which,beingpseudoscalars,
scription in terms of an underlying fundamental quantum
are much easier to handle. Although this is a high-energy
field theory is out of reach. The two cases of interest for
limit, there is a generalization to the effective Lagrangian
this work are, on the one hand, chiral perturbation theory
formalism [4] that, for practical purposes, allows us to
(ChPT) [1] which describes effectively the low-energy
identify, up to the difference in scales, the formalisms of
dynamics of hadrons, inaccessible to perturbative QCD
SU(cid:1)2(cid:2)ChPTandtheSISBS,andtherefore,fromnowonwe
calculations in terms of quarks and gluons; and, on the
will be referring to ChPT, but keeping in mind that our
other hand, the effective description of the strongly inter-
results have a straightforward translation to the SISBS.
actingelectroweaksymmetrybreakingsector(SISBS)[2],
Both cases above are examples of strongly interacting
whoseunderlying fundamental theory remains unknown.
Both cases have in common the existence of a sponta- systems whose most salient feature is the saturation of
neous symmetry breaking of a global chiral SU(cid:1)N(cid:2) (cid:3) unitarity and the associated resonant states, which lie
L
SU(cid:1)N(cid:2) group down to an SU(cid:1)N(cid:2) group. The beyond the reach of perturbative energy expansions.
R L(cid:4)R
Goldstone theorem implies the presence of N2(cid:5)1 mass- Thus, it may seem that the use of effective Lagrangians
lessGoldstonebosons(GB)intheparticlespectrum.These is limited to energies below those resonances, whose ef-
GB thus become the relevant degrees of freedom of the fects are encoded in the values of higher order effective
system below a chiral scale (cid:1) , where an effective chiral coupling constants. However, since unitarity fixes the
(cid:1)
Lagrangian can be built in terms of just those GB as the imaginary part of inverse partial waves in the elastic re-
most general derivative expansion consistent with the gion, the effective Lagrangian approach is also useful in
knownsymmetries.Notethatthereisawell-definedpower the resonant region, for instance, used inside a dispersion
counting so that divergences generated in loop diagrams relation,inordertoobtaintherestoftheamplitude.These
that use vertices up to a given order can be renormalized techniquesareknownasunitarizationmethods,andrepro-
into the coefficients of higher order terms. In this sense duce simultaneously the low-energy expansion and the
these effective theories are renormalizable order by order. lightest resonances without including them explicitly in
In addition, in both examples, there are small explicit the Lagrangian. The great advantage is that such reso-
symmetry breaking terms. For QCD the relatively nances and their properties are generated without prejudi-
small masses of the lightest three quarks provide a mass ces about their nature or their existence. Also, since the
M (cid:6)(cid:1) to the GB, identified with the pions, kaons, and Lagrangian symmetries andsomefeatures ofthe effective
(cid:1)
etas,sothattheeffectiveapproachbecomes,inpractice, a constants can be directly related to the underlying theory,
derivative and mass expansion. For the SISBS, there is a like QCD, one can study the properties of these states
local SU(cid:1)2(cid:2) (cid:3)U(cid:1)1(cid:2) symmetry whose gauge bosons based on more fundamental grounds. One of the most
L
couple to the ‘‘would-be GB’’ that, in a unitary gauge, extensively used unitarization techniques is the inverse
disappear from the spectrum, giving rise to gauge boson amplitude method (IAM) [5–8], which uses the fully re-
longitudinal components that thus acquire a mass M . In normalizedeffectivechiralexpansion,withoutanyfurther
V
thiswaytheSU(cid:1)2(cid:2) (cid:3)U(cid:1)1(cid:2)gaugesymmetryisspontane- approximationandwithoutintroducinganyotherspurious
L
ously, but not explicitly, broken to the electromagnetic parameter, but just the effective constants up to a given
group U(cid:1)1(cid:2) . In this case, in addition to the derivative order. Within hadronic physics, it generates the well-
EM
expansion, one expands also in terms of electroweak cou- known vector resonances and the more controversial sca-
1550-7998=2008=77(5)=056006(13) 056006-1 © 2008 The American Physical Society
GO´MEZ NICOLA,PELA´EZ, ANDR´IOS PHYSICALREVIEW D77, 056006 (2008)
lars using parameters consistent with one-loop ChPT, al-
lowing one to establish their different nature in terms of
their dependence on the number of colors. The IAM has
also been extended to two loops [7,9], the finite tempera-
ture formalism [10,11], and to the pion-nucleon sector
[12]. Within the SISBS [13], it provides the prediction of
the general resonance spectrum and how well it could be
detected at the CERN LHC.
However,itisknown[7,14]thattheIAMfailstorepro-
ducecorrectly theAdlerzerosthatappear inthesubthres-
hold region of some partial waves as a consequence of
chiral symmetry [15]. Furthermore, it generates spurious
poles, or ‘‘ghosts,’’ thus questioning its reliability in that
region,and alsocasting some doubtsabout the robustness
oftheresultsinthephysicalregionifsuchstructureswere
properly accounted for. FIG. 1. Analyticstructureofpion-pionscatteringpartialwaves
The aim of this paper is to show that a very simple and the integration contour used to obtain their dispersion
modification of the IAM can correctly take into account relations.
those zeros and ghosts. This modification corresponds to
terms that had been neglected in the original dispersive The inverse of t(cid:1)s(cid:2) has the same analytic structure,
derivation of the IAM since they contribute to higher except for the possible presence of poles corresponding
orders in the chiral expansion. Actually, we will check tozerosoft(cid:1)s(cid:2).Inparticular,chiralsymmetryrequiresthe
explicitly that sucha procedure is justified in the physical existence of the so-called Adler zeros below threshold in
region, where these modifications yield negligible contri- scalarpartialwaves[1,15],thecaseofinterestinthiswork,
butions,thusshowingtherobustnessofthestandardIAM. which we denote by s . Hence it is then possible towrite
A
However,apartfromimprovingtheIAMconsistency,these thefollowingdispersionrelationfortheinverseamplitude:
terms are essential in the subthreshold region which is
1 1 s(cid:5)z Z1 Im1=t(cid:1)z(cid:2)
relevant to study the effect of chiral symmetry restoration (cid:7) (cid:4) 0 dz (cid:4)LC(cid:1)1=t(cid:2)
t(cid:1)s(cid:2) t(cid:1)z (cid:2) (cid:3) (cid:1)z(cid:5)s(cid:2)(cid:1)z(cid:5)z (cid:2)
on resonances [11], or their dependence on quark masses 0 sth 0
[16]. (cid:4)PC(cid:1)1=t(cid:2): (1)
InthenextsectionwewillthusrevisitthestandardIAM
Hereandinthefollowing,wewillsimplywritezinsteadof
derivation from dispersive theory, where Adler zeros are
z(cid:4)i(cid:4) with (cid:4)>0 for the imaginary parts inside the cut
neglected, paying special attention to the role of those
integrals. Note that wehave explicitly written the integral
zerosinthesubthresholdregion.InSec.IIIwewillpresent
over the right-hand cut (or physical cut, extending from
anaivewayofextendingtheIAMamplitude,withoutusing
thresholds toinfinity),butwehaveshortenedbyLCthe
dispersion relations, that solves the caveats in that region. th
equivalent expression for the left cut (from (cid:5)1 to 0) and
Section IV will show a dispersive derivation of a more
the pole contribution. We could proceed in the same way
general modified amplitude, for the case of equal masses
withothercuts,ifpresent,asinthe(cid:3)Kcase.Inadditionwe
(e.g. (cid:3)(cid:3) scattering). The case of unequal masses, like in
have made one subtraction to ensure convergence, at a
(cid:3)K scattering, deserves a separate discussion, for the
point z (cid:1)s ;s .
reasons explained in Sec. V. Finally, in Sec. VI we will 0 A 2
We now recall that unitarity, for physical values of s in
present some numerical results for the modified
the elastic region, implies
amplitudes.
1
Imt(cid:1)s(cid:2)(cid:7)(cid:5)(cid:1)s(cid:2)jt(cid:1)s(cid:2)j2 )Im (cid:7)(cid:5)(cid:5)(cid:1)s(cid:2); (2)
II.THE INVERSEAMPLITUDEMETHOD t(cid:1)s(cid:2)
p
A.Dispersive derivation where(cid:5)(cid:1)s(cid:2)(cid:7)2p = (cid:1)s(cid:1)(cid:1).LetusremarkthatsinceIm1=t(cid:7)
CM
Theone-channelIAM[5–7,9]canbeobtainedbyusing (cid:5)(cid:5)we knowexactly the integrand over the elastic cut.
ChPT up to a given order inside a dispersion relation. To In contrast, ChPT amplitudes are obtained as a series
simplify the discussion, let us first consider pion-pion expansion t(cid:1)s(cid:2)(cid:7)t2(cid:1)s(cid:2)(cid:4)t4(cid:1)s(cid:2)(cid:4)... where t2(cid:1)s(cid:2)(cid:7)O(cid:1)p2(cid:2),
scattering, partial wave amplitudes of definite isospin I t4(cid:1)s(cid:2)(cid:7)O(cid:1)p4(cid:2),and p standsforthe pion mass or momen-
and angular momentum J, although for brevity we will tum.Thereforeelasticunitarityisnotsatisfiedexactly,but
simplycall them t,whoseanalytic structure inthe splane only order by order as follows:
isshowninFig.1.Thephysicalright-handcutcomesfrom Imt (cid:1)s(cid:2)(cid:7)0; Imt (cid:1)s(cid:2)(cid:7)(cid:5)(cid:1)s(cid:2)jt (cid:1)s(cid:2)j2;...: (3)
2 4 2
unitarityandstartsatthresholds ,whiletheleft-handcut
th
comes from the t,u channels. Letusalsonotethatt (cid:1)s(cid:2)isapurepolynomialandhasno
2
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INVERSE AMPLITUDEMETHODANDADLERZEROS PHYSICALREVIEW D77, 056006 (2008)
cuts,andwecanthuswriteatrivialdispersionrelationfor contribution to t(cid:1)s(cid:2), even without its explicit calculation,
1=t (cid:1)s(cid:2) that reads hastobeO(cid:1)p6(cid:2).InSecs.III,IV,andVwewillcalculateit
2
explicitly to arrive at the modified IAM.
1 1
(cid:7) (cid:4)PC(cid:1)1=t (cid:2); (4)
t (cid:1)s(cid:2) t (cid:1)z (cid:2) 2
2 2 0 B. IAMproperties andits naive derivation
wherenowthepolecontributionisduetos2,theAdlerzero Incidentally,we can recast Eq. (2) as
of t . In addition, except for the poles, the function
2
t (cid:1)s(cid:2)=t2(cid:1)s(cid:2) has the same analytic cut structure of 1=t, and, 1
4 2 t(cid:1)s(cid:2)(cid:7) ; (9)
using Eqs. (2) and (3), over the physical cut wefind Ret(cid:5)1(cid:1)s(cid:2)(cid:5)i(cid:5)(cid:1)s(cid:2)
t (cid:1)s(cid:2) 1 and thus it seems that the IAM can also be derived in a
Im 4 (cid:7)(cid:5)(cid:1)s(cid:2)(cid:7)(cid:5)Im : (5)
t2(cid:1)s(cid:2) t(cid:1)s(cid:2) much simpler way by replacing Ret(cid:5)1 by its O(cid:1)p4(cid:2) ChPT
2 expansionRet(cid:5)1 (cid:7)(cid:1)t (cid:5)Ret (cid:2)=t2.Thisisthewayunitar-
2 4 2
Wecanthereforewriteanotherdispersionrelationsimi- ization methods are usually presented, although it makes
lar to that of 1=t(cid:1)s(cid:2), but fort4(cid:1)s(cid:2)=t22(cid:1)s(cid:2), no use of the strong analytic constraints of amplitudes,
whichareindeed absentinEq.(9).Furthermore,the criti-
t (cid:1)s(cid:2) t (cid:1)z (cid:2) s(cid:5)z Z1 Imt (cid:1)z(cid:2)=t (cid:1)z(cid:2)2
4 (cid:7) 4 0 (cid:4) 0 dz 4 2 cismisimmediatelyraisedthattheChPTexpansioncannot
t2(cid:1)s(cid:2)2 t2(cid:1)z0(cid:2)2 (cid:3) sth (cid:1)z(cid:5)s(cid:2)(cid:1)z(cid:5)z0(cid:2) be usedat high energies.
(cid:4)LC(cid:1)t =t2(cid:2)(cid:4)PC(cid:1)t =t2(cid:2); (6) However, note that the dispersive one-channel IAM
4 2 4 2
derivation in the previous section imposes analyticity in
wherethepolecontribution,onceagain,isduetotheAdler the form of two dispersive integrals and makes use of the
zero of t . ChPT expansion (up to one loop in this case) for the
2
We are now going to relate the dispersion relation for subtraction constants and the left cut. The use of ChPT
1=t(cid:1)s(cid:2)withthatfort (cid:1)s(cid:2)=t2(cid:1)s(cid:2).Aswealreadycommented, for the subtraction constants is well justified, since it is
4 2
Im1=t(cid:1)s(cid:2)(cid:7)(cid:5)Imt =t2(cid:1)s(cid:2)ontherightcut,andthereforethe usedats(cid:7)z inthelow-energyregion.Sincetheintegral
4 2 0
integrals over the physical cuts for 1=t(cid:1)s(cid:2) and t =t2(cid:1)s(cid:2) are extendstoinfinity,ChPTmayseemaworseapproximation
4 2
exactlyoppositetoeachother.Inaddition,usingChPTwe for the left cut or possible inelastic cuts; however, the
find that LC(cid:1)1=t(cid:2)’(cid:5)LC(cid:1)t =t2(cid:2), which is a well-justified subtractionsuppressesthehigh-energyregionthatcontrib-
4 2
approximation,since,duetothesubtraction,theintegrand uteslittle,asexplainedabove.Furthermore,whentheIAM
of LC is weighted at low energies, precisely where ChPT is usedfor physical values of s above the physical thresh-
applies. Finally, we have to evaluate the subtraction con- old, the left cut is damped by an additional 1=(cid:1)z(cid:5)s(cid:2)
stantinEq.(1),andthiscanonlybedoneaslongasz isin factor, and not only the high-energy part, but its whole
0
thelow-energyregion,whereitisperfectlyjustifiedtouse contributionisrathersmall.Therearenomodeldependent
ChPTtofind1=t(cid:1)z (cid:2)’1=t (cid:1)z (cid:2)(cid:5)t (cid:1)z (cid:2)=t (cid:1)z (cid:2)2.However, assumptions,butjustapproximationstoagivenorder,and
0 2 0 4 0 2 0
notethatthisexpansionisaverybadapproximationforz therefore the approach provides an elastic amplitude that
0
near s or s , where t and t vanish. Therefore, we only satisfiesunitarityandhasthecorrectChPTexpansionupto
2 A 2
know how to relate those dispersion relations for subtrac- that given order. It is also straightforward to extend it to
tionpointsz inthelow-energyregion,butsufficientlyfar higher orders[7,9].
0
from the Adler zeros. In Sec. VI we will check that the Moreover, Eq. (2) is only valid on the physical cut,
resultshaveverylittlesensitivitytothechoiceofz aslong whereas the dispersive derivation allows us to consider
0
asitliesinthisregion.Whenthisisthecase,usingEqs.(1), the amplitude in the complex plane and, in particular, to
(4), and (6) we can write 1=tas look for poles of the associated resonances. Actually, al-
ready tenyears ago [7] the poles for the (cid:6)(cid:1)770(cid:2), K(cid:8)(cid:1)892(cid:2),
1 1 t (cid:1)s(cid:2)
’ (cid:5) 4 (cid:5)PC(cid:1)1=t (cid:2)(cid:4)PC(cid:1)t =t2(cid:2)(cid:4)PC(cid:1)1=t(cid:2): and most interestingly, the controversial (cid:5) [also called
t(cid:1)s(cid:2) t2(cid:1)s(cid:2) t2(cid:1)s(cid:2)2 2 4 2 f0(cid:1)600(cid:2)] were generated without any model dependent
(7) assumptions.
Obviously, and contrary to wide belief, the IAM con-
The standard dispersive derivation of the IAM [6,7]
tains a left cut and respects crossing symmetry up to, of
simplyneglectedthesumofpolecontributionstoarriveat
course,theorderintheChPTexpansionthathasbeenused.
t2(cid:1)s(cid:2) The confusion may come from the fact that the IAM has
tIAM(cid:1)s(cid:2)’ 2 ; (8) also been applied in a coupled channel formalism, for
t (cid:1)s(cid:2)(cid:5)t (cid:1)s(cid:2)
2 4 which there is still no dispersive derivation, and has been
thus providing an elastic amplitude that satisfies unitarity frequently used by approximating further the amplitudes
and has the correct low-energy expansion of ChPT up to neglecting tadpole and left cut terms [17]. But, strictly
the order we have used. When such amplitude is chirally speaking,thatwouldnotbethefullIAM,whichdefinitely
expanded to O(cid:1)p4(cid:2), this implies [7] that the total pole has a left cut.
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GO´MEZ NICOLA,PELA´EZ, ANDR´IOS PHYSICALREVIEW D77, 056006 (2008)
C.IAMcaveats:Adler zeros andghosts theappearance ofa spuriouspole onthe real axisclose to
the Adler zero. As a consequence, the predictions of the
To end this section, we recall that in the dispersive
standard IAM below threshold and, in particular, around
derivation the sum of pole terms (PC) in Eq. (7) is ne-
the Adler zero position are not reliable.
glected, since it yields a higher order contribution [7].
In fact, note that, since the interval s2(cid:1)0;s (cid:2) lies
However, this leads to a couple of problems related to th
within the convergence region of the chiral expansion
the presence of the Adler zeros below threshold in the
and t changes sign at s(cid:7)s , t (cid:5)t turns out to be, for
scalar waves. Let us first note that, despite we have used 2 2 2 4
the cases of interest here, a continuous, monotonically
intheIAMtheChPTexpansionuptonexttoleadingorder
(NLO),duetothet (cid:1)s(cid:2)2factorinthenumerator,itvanishes increasing (or decreasing) function in (cid:1)0;sth(cid:2) that changes
2
sign from s(cid:7)0 to s(cid:7)s . Therefore, there is one single
ats ,whichisonlytheleadingorder(LO)chiralapproxi- th
2
point s~where the denominator of (8) vanishes, which, as
mationtotheexactAdlerzeros .Inaddition,itisadouble
A
long as t (cid:1)s (cid:2)(cid:1)0, produces the spurious pole below
zeroinsteadofasinglezero.This,aswewillsee,leadsto 4 2
I 0, J 0 I 0, J 0
0.04
1
0.03
0.8 0.02
0.01
0.6 0 sA
0.4 0.01 s2
tIAM 0.02 tIAM
0.2 tmIAM 0.03 tmIAM
200 400 600 800 1000 70 80 90 100 110 120 130 140
s MeV s MeV
I 2, J 0 I 2, J 0
0.5 0.004
0.003
0.4
0.002
0.3 0.001
sA
0
0.2 0.001 s2
tIAM 0.002 tIAM
0.1
tmIAM 0.003 tmIAM
200 400 600 800 1000 195 196 197 198 199 200
s MeV s MeV
I 3 2, J 0 I 3 2, J 0
0.4 0.2
0.35 0.15
0.3 0.1
0.25 0.05
0.2 0 sA
s2
0.15 0.05
0.1 tIAM 0.1 tIAM
0.05 tmIAM 0.15 tmIAM
400 500 600 700 800 900 1000 480 490 500 510 520 530 540 550
s MeV s MeV
I 1 2, J 0 I 1 2, J 0
0.2
1.2
0.15
1 0.1
0.8 0.05
0.6 0 sA s2
0.05
0.4
tIAM 0.1 tIAM
0.2 tmIAM 0.15 tmIAM
400 500 600 700 800 900 1000 475 480 485 490 495 500 505 510
s MeV s MeV
FIG. 2. ComparisonbetweentheIAMandthemIAMfordifferentisospinIpartialwavesof(cid:3)(cid:3)and(cid:3)KinthescalarJ(cid:7)0channel.
Theleftcolumncoverstheregionfromtheleftcutupto1GeV.Theonlysignificantdifferencesbetweenbothmethodsoccursinthe
region around the Adler zeros of each partial wave, which is shown in detail in the right column.
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INVERSE AMPLITUDEMETHODANDADLERZEROS PHYSICALREVIEW D77, 056006 (2008)
threshold discussed above, i.e., a nonobserved (cid:3)(cid:3) bound approximationstotheAdlerzeroatLO,s ,andNLO,s (cid:4)
2 2
state. s .Thus, t (cid:1)s (cid:2)(cid:7)0 and t (cid:1)s (cid:4)s (cid:2)(cid:4)t (cid:1)s (cid:4)s (cid:2)(cid:7)0.
4 2 2 2 2 4 4 2 4
For instance, let us consider I (cid:7)J (cid:7)0 (cid:3)(cid:3) scattering In this section we present a naive, and intuitive, deriva-
(for the values of the low-energy constants quoted in tion leading to a partial wave definition which does not
Sec. VI). This is an attractive channel, so that t (cid:5)t is havetheAdlerzerorelatedproblemsdiscussedabove.The
2 4
positiveatthreshold.Sincethatfunctionisnegativeats(cid:7) derivation follows closely [11], where this problem was
0andats(cid:7)s ,s~>s inthat case,asshownintheupper addressed in the context of real pion scattering poles aris-
2 2
right panel of Fig. 2, we find s~’(cid:1)110 MeV(cid:2)2 and s ’ ingfrom medium effects suchastemperature anddensity.
2
(cid:1)99 MeV(cid:2)2. In the I (cid:7)2, J (cid:7)0 channel, which is repul- In that paper, there was not a formal proof based on
sive, t (cid:5)t is positive at s(cid:7)0 and negative at s(cid:7)s dispersion relations, such as the one we will present here
2 4 2
[t (cid:1)s (cid:2)>0] so that s~<s . In that channel, s~’ later, and it waslimited to pion-pion scattering.
4 2 2
(cid:1)196 MeV(cid:2)2 and s ’(cid:1)198 MeV(cid:2)2 as seen also in Fig. 2. From the discussion in the previous sections, it is clear
2
Letusalsopointoutthat,togetherwiththefirstRiemann that if we modify the inverse amplitude as 1=tIAM(cid:1)s(cid:2)!
sheetspuriouspolejustdiscussed,theIAMhasacompan- 1=tIAM(cid:1)s(cid:2)(cid:4)A(cid:1)s(cid:2)=t2 withA(cid:1)s(cid:2)ananalyticfunctionatleast
2
ionpoleinthesecondRiemannsheetbelowthreshold.For off the real axis, real for real s, the unitarity and analytic
instance, the second-sheet IAMfor(cid:3)(cid:3) scattering reads propertiesoftheamplituderemainunaltered.Themodified
IAM(from nowon called mIAM),then reads
t (cid:1)s(cid:2)2
t(cid:1)s(cid:2)IAM;IIsheet (cid:7) 2 (10) t2(cid:1)s(cid:2)
t (cid:1)s(cid:2)(cid:5)t (cid:1)s(cid:2)(cid:5)2(cid:5)~(cid:1)s(cid:2)t2(cid:1)s(cid:2) tmIAM(cid:1)s(cid:2)(cid:7) 2 : (11)
2 4 2 t (cid:1)s(cid:2)(cid:5)t (cid:1)s(cid:2)(cid:4)AmIAM(cid:1)s(cid:2)
2 4
p(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
where (cid:5)~(cid:1)s(cid:2)(cid:7)i(cid:5)(cid:1)s(cid:5)i0(cid:4)(cid:2)(cid:7) 4m2=s(cid:5)1 for 0<s< Considernowthecaseof(cid:3)(cid:3)scattering,wherewehave
(cid:3)
4m2. Thus, if we are dealing with an attractive channel, simplyt (cid:1)s(cid:2)(cid:7)t0(cid:1)s (cid:2)(cid:1)s(cid:5)s (cid:2)witht0(cid:1)s (cid:2)constant.Thenthe
(cid:3) 2 2 2 2 2 2
like the 00 one, the denominator of (10) is positive near Laurent expansion around s(cid:7)s of the standard IAM
2
threshold (dominated by t >0) and diverges to minus reads
2
infinity as s!0(cid:4), so that it must have at least one zero,
which again generates a pole if t4(cid:1)s2(cid:2)(cid:1)0. Since (cid:5)(cid:5)~t22 in 1 (cid:7)(cid:5) t4(cid:1)s2(cid:2) (cid:4)t02(cid:1)s2(cid:2)(cid:5)t04(cid:1)s2(cid:2)
Eq. (10) is also an increasing function, there will be only tIAM(cid:1)s(cid:2) t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2)2 t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2)
2 2 2 2 2 2
one suchpole. (cid:4)O(cid:1)s(cid:5)s (cid:2)0: (12)
2
In conclusion, the existence of the perturbative Adler
zeros and the fact that the IAMamplitude doesnot repro- The idea is that if we want the amplitude to have only an
duce them well leads to the presence of spuriouspoles. A Adler zero of order one at s(cid:7)s , we must subtract from
A
similar conclusion had been noticed in [18]. In the next 1=tIAM the above double and single pole contributions at
sections, we present, first, a very simple construction of a s(cid:7)s and add a single pole at s , i.e.,
2 A
modifiedIAM,alongthe linesofthepreviouslydiscussed
naive derivation of the standard IAM obtained without 1 1 t (cid:1)s (cid:2) t0(cid:1)s (cid:2)(cid:5)t0(cid:1)s (cid:2)
(cid:7) (cid:4) 4 2 (cid:5) 2 2 4 2
using dispersion relations, which solves these problems, tmIAM(cid:1)s(cid:2) tIAM(cid:1)s(cid:2) t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2)2 t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2)
2 2 2 2 2 2
and next we show two dispersive derivations of the modi-
c
fiedIAM.Oneofthemissubtractedatarbitraryz (within (cid:4) ; (13)
0 s(cid:5)s
the range of validity of our approximations) and the other A
oneattheAdlerzero.WewillshowhowthemodifiedIAM
where cis a so far undetermined constant that, as wewill
obtained naively corresponds to a particular limit of the
show now, can be fixed by demanding that the mIAM
first dispersive approach and comes out directly in the
formula matches the perturbative ChPT series to fourth
second. Finally, we will also show that the differences order,namely, A(cid:7)O(cid:1)p6(cid:2).
betweenthemodifiedIAMformulasarenegligiblenumeri-
In practice, it is simpler to keep track of the different
cally,andthat,whilefixingtheabove-mentionedproblems, chiralpowersbycountingthepowersoff(cid:5)2,wherefisthe
themodifiedIAMdoesnotyieldanysignificantmodifica- pion decay constant. Thus, since t0(cid:1)s (cid:2)(cid:7)O(cid:1)f(cid:5)2(cid:2), t (cid:7)
2 2 4
tionoverthestandardIAMinthephysicalregionandtothe O(cid:1)f(cid:5)4(cid:2), and s (cid:7)O(cid:1)f(cid:5)2(cid:2), expanding the expression
4
resonance poles. Therefore, the results obtained so far in
t (cid:1)s (cid:4)s (cid:2)(cid:4)t (cid:1)s (cid:4)s (cid:2)(cid:7)0around s we find
2 2 4 4 2 4 2
the literaturewith the IAMremainvalid.
s (cid:7)(cid:5)t (cid:1)s (cid:2)=t0(cid:1)s (cid:2)(cid:4)O(cid:1)f(cid:5)4(cid:2): (14)
4 4 2 2 2
III.MODIFIEDIAM:NAIVEDERIVATION
Using this in Eq. (13) with s (cid:7)s (cid:4)s (cid:4)O(cid:1)f(cid:5)4(cid:2), and
A 2 4
First of all, we will set some notation: as before, we requiringthatEq.(11)matchesthechiralexpansionatlow
denote by s the Adler zero of the ‘‘complete’’ partial energies,wefindthefirsttwoordersofthechiralexpansion
A
wave, namely, t(cid:1)s (cid:2)(cid:7)0. In addition, wewill also use the forc:
A
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TABLE I. (cid:5)and(cid:7)polepositionscalculatedwiththeIAM,the 1 (cid:2) 1 1 (cid:3)
mIAM,and the z0IAMwith z0 (cid:7)sth. PC(cid:1)1=t2(cid:2)(cid:7)t0(cid:1)s (cid:2) s(cid:5)s (cid:5)z (cid:5)s ; (17)
2 2 2 0 2
Method (cid:5)pole (cid:7)pole
t (cid:1)s (cid:2) (cid:2) 1 1 (cid:3)
IAM 443:71(cid:5)i217:58 724:2(cid:5)i216:2 PC(cid:1)t =t2(cid:2)(cid:7) 4 2 (cid:5)
mIAM 443:68(cid:5)i217:56 725:3(cid:5)i216:3 4 2 t0(cid:1)s (cid:2)2 (cid:1)s(cid:5)s (cid:2)2 (cid:1)z (cid:5)s (cid:2)2
2 2 2 0 2
z0IAM, z0 (cid:7)sth 443:82(cid:5)i216:99 727:7(cid:5)i210:0 t0(cid:1)s (cid:2) (cid:2) 1 1 (cid:3)
(cid:4) 4 2 (cid:5) ; (18)
t0(cid:1)s (cid:2)2 s(cid:5)s z (cid:5)s
2 2 2 0 2
1 t0(cid:1)s (cid:2)
c(cid:7) (cid:5) 4 2 (cid:4)O(cid:1)f(cid:5)2(cid:2); (15) 1 (cid:2) 1 1 (cid:3)
t0(cid:1)s (cid:2) t0(cid:1)s (cid:2)2 PC(cid:1)1=t(cid:2)(cid:7) (cid:5) ; (19)
2 2 2 2 t0(cid:1)s (cid:2) s(cid:5)s z (cid:5)s
A A 0 A
which leads to where we have assumed a single zero in t(cid:1)s (cid:2), as the
A
presence of the 1=(cid:1)s(cid:5)s (cid:2) factor in PC(cid:1)1=t(cid:2) shows.
(cid:1)s (cid:5)s (cid:2)(cid:1)s(cid:5)s (cid:2) A
AmIAM(cid:1)s(cid:2)(cid:7)t (cid:1)s (cid:2)(cid:5) 2 A 2 (cid:9)t0(cid:1)s (cid:2)(cid:5)t0(cid:1)s (cid:2)(cid:10): As we have discussed above, and as is detailed in
4 2 s(cid:5)sA 2 2 4 2 the Appendix, by expanding chirally (cid:5)PC(cid:1)1=t (cid:2)(cid:4)
2
(16) PC(cid:1)t =t2(cid:2)(cid:4)PC(cid:1)1=t(cid:2),thepolescontributetot(cid:1)s(cid:2)athigher
4 2
order, and that is why they were customarily neglected.
Therefore, the mIAM in Eq. (11) with AmIAM(cid:1)s(cid:2) in
However, these pole contributions contain the terms
Eq.(16)ands approximatedbyitschiralexpansiongiven
A needed to have the Adler zero in the correct position. In
abovematchesthechiralexpansionoftheamplitudeupto
addition, by taking the limit s!s , (cid:1)(cid:5)PC(cid:1)1=t (cid:2)(cid:4)
fourthorderandhastheAdlerzeroatthesamepositionand 2 2
PC(cid:1)t =t2(cid:2)(cid:2) tends to the term needed to cancel the double
with the same order as the perturbative amplitude. 4 2
zerooftheIAMins [seeEq.(12)],andthespuriouspole
Furthermore, we have solved, in turn, the spurious pole 2
will also disappear.
problem. In fact, since AmIAM(cid:1)s (cid:2)(cid:7)t (cid:1)s (cid:2), the denomina-
2 4 2 Insummary,themodifiedIAMobtainedfromdispersive
torofEq.(11)vanishesats(cid:7)s .But,fors(cid:1)s wehave
2 2 relations subtracted at z , that we will denote by z IAM,
shownthatAmIAM(cid:1)s(cid:2)(cid:7)O(cid:1)f(cid:5)6(cid:2),andthereforeourprevious 0 0
can bewritten again as
argument about the monotonous behavior of the denomi-
nator still holds so that the denominator vanishes only at t2(cid:1)s(cid:2)
s(cid:7)s2, where there is no pole contribution. The same tz0IAM(cid:1)s(cid:2)’t2(cid:1)s(cid:2)(cid:5)t4(cid:1)s2(cid:2)(cid:4)Az0IAM(cid:1)s(cid:2); (20)
holds for the spurious second-sheet poles. In Fig. 2 we
show the mIAM amplitude in the I (cid:7)J (cid:7)0 channel, and where now
we observe that is not singular below threshold and re-
t (cid:1)s(cid:2)2
mAdailnerszcelrooseretgoiotnh.eTshtaensdaamrdeIsAitMuatiroensutlatkaewsapylacfreomin tthhee Az0IAM(cid:1)s(cid:2)(cid:7)AmIAM(cid:1)s(cid:2)(cid:5)t22(cid:1)z0(cid:2)2AmIAM(cid:1)z0(cid:2): (21)
I (cid:7)2, J (cid:7)0 channel. Finally, we have checked that, as
Of course, the position of the Adler zero for the total
expected from our previous arguments, the f (cid:1)600(cid:2) or (cid:5)
0 amplitude is not known, but since we have been working
pole remains at the same place either using the second
withChPTtooneloop,wecanimposetheAdlerzerotobe
Riemann sheet extensions of the mIAM or the IAM (see
located in its one-loop position; namely, in the above
Table I).
formulas, we have to replace s !s (cid:4)s which is ob-
A 2 4
Inthenextsections,wewillcheckhowamodifiedIAM
tained from the equation t (cid:1)s (cid:4)s (cid:2)(cid:4)t (cid:1)s (cid:4)s (cid:2)(cid:7)0 as
2 2 4 4 2 4
can also be obtained by considering explicitly the pole
explained in Sec. III. Thus, to obtain (21) we have made
contributions in the dispersive derivation, thus ensuring use of t (cid:1)s(cid:2)(cid:7)t0(cid:1)s (cid:2)(cid:1)s(cid:5)s (cid:2) and chirally expanded
2 2 2 2
the correct analytic properties of the amplitude. We will 1=t0(cid:1)s (cid:2)’1=t0(cid:1)s (cid:2)(cid:5)t0(cid:1)s (cid:2)=t0(cid:1)s (cid:2)2,whichisperfectlyjus-
also show that the modified formula obtained with the A 2 2 4 2 2 2
tified near s . Note that we can use the chiral expansions
A
naive derivation in this section can also be obtained as a
arounds ands becausewenolongerexpandtheinverse
A 2
particular case.
oftheamplitudesbutratherthatoftheirderivatives,asthey
appear in the residues of the pole contributions.
IV. MODIFIEDIAM: DISPERSIVEDERIVATION Notethat,onceagain,thefactorAmIAM(cid:1)s(cid:2)thatappeared
FOR EQUALMASSES in the previous section is present, but now there is an
additional and very similar piece that carries a z depen-
0
A.Pole contributionto the standardderivation
dence, which occurs due to our truncation of the ChPT
The derivation of the modified IAM from dispersion series when approximating the subtraction constants and
theory follows that in Sec. II up to Eq. (7), but keeping pole contributions. This additional term in Az0IAM is also
the pole contributions, which, by evaluating the corre- O(cid:1)f(cid:5)6(cid:2),but,aswewillseebelow,aslongasz lieswithin
0
sponding residues, read the range where our approximations remain valid, it is
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numerically small not only in the physical region but also Outsidethatregion,theothertermsbecomerelevantand
below threshold. Therefore, this term can be dropped thedifferencewithourpreviousderivationsisthatnowwe
withoutspoilingtherightchiralbehavioroftheamplitude alsoapproximatethe1=tintegraltermovertherightcutby
in the subthreshold region, obtaining again the mIAM in using (cid:1)s(cid:5)s (cid:2)=(cid:1)z(cid:5)s (cid:2)’(cid:1)s(cid:5)s (cid:2)=(cid:1)z(cid:5)s (cid:2), which is its
A A 2 2
Sec. III, which, for the moment, we have justified only LO chiral expansion. This is a remarkably good approxi-
numerically. mationforthedispersionrelationaslongaszissufficiently
Furthermore, it is tempting to take the z !1 limit in farfroms ands ,whichisindeedthecasefortherightcut
0 2 A
Eq.(21)andrecoverthemIAMoftheprevioussectionby integral.Ofcourse,the1=trightcuttermshouldvanishat
noting that the second term in Eq. (21) vanishes in that s andnowitdoesnot,but,aswehavejustcommented,the
A
limit. However, this is just a formal justification of our pole contribution diverges precisely at s and thus is
A
naivederivationsincewerequiredthesubtractionpointz largely dominant over the integral, which therefore can
0
to remain in the low-energy applicability region of ChPT. be completely neglected.
Nevertheless,wewillseeinthenextsectionthatthereisan Onceagain,wesimplyaddEqs.(22)and(23)toobtain
alternative dispersive derivation of the mIAM, which is the mIAMequation
somewhatdifferentfromthestandarddispersivederivation
1 t (cid:1)s(cid:2) t (cid:1)s (cid:2) t0(cid:1)s (cid:2)
since it requires subtractions at the Adler zero of each (cid:7)(cid:5) 4 (cid:4) 4 2 (cid:4) 4 2
tmIAM(cid:1)s(cid:2) t (cid:1)s(cid:2)2 t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2)2 t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2)
function. 2 2 2 2 2 2 2
1
(cid:4) ; (26)
B. Subtractionat the Adler zeros t0(cid:1)s (cid:2)(cid:1)s(cid:5)s (cid:2)
A A
Theway to derive Eq. (16) from dispersive theory is to where, in the pole contributions, wehave usedthat
makethesubtractionsattheplacewherewealreadyhavea
t00(cid:1)s (cid:2) t00(cid:1)s (cid:2)
pole.Note,however,that1=thasitspoleintheAdlerzero (cid:5) A (cid:4) 4 2 (cid:7)O(cid:1)f(cid:5)2(cid:2); (27)
at sA whereas t4=t22 has its pole at s2, so that we have to 2t0(cid:1)sA(cid:2)2 2t02(cid:1)s2(cid:2)2
write
which, once again, can be safely neglected since it corre-
1 s(cid:5)s Z (cid:5)(cid:1)z(cid:2) spondto the chiral expansion at very lowenergies.
(cid:7)(cid:5) A dz (cid:4)LC(cid:1)1=t(cid:2) Finally,if we evaluate perturbatively
t(cid:1)s(cid:2) (cid:3) (cid:1)z(cid:5)s (cid:2)(cid:1)z(cid:5)s(cid:2)
RC A
(cid:4)PC(cid:1)1=t(cid:2); (22) 1 ’ 1 (cid:5) t04(cid:1)s2(cid:2) (cid:4)O(cid:1)f(cid:5)2(cid:2); (28)
t0(cid:1)s (cid:2) t0(cid:1)s (cid:2) t0(cid:1)s (cid:2)2
A 2 2 2 2
t4(cid:1)s(cid:2) (cid:7)s(cid:5)s2 Z dz (cid:5)(cid:1)z(cid:2) (cid:4)LC(cid:1)t =t2(cid:2) and add 0 to Eq. (26) written as 0(cid:7)1=t2(cid:1)s(cid:2)(cid:5)
t2(cid:1)s(cid:2)2 (cid:3) RC (cid:1)z(cid:5)s2(cid:2)(cid:1)z(cid:5)s(cid:2) 4 2 1=(cid:1)t02(cid:1)s2(cid:2)(cid:1)s(cid:5)s2(cid:2)(cid:2), we reobtain Eq. (11) with A(cid:1)s(cid:2) given
(cid:4)PC(cid:1)t =t2(cid:2); (23) by Eq. (16); i.e., we recover our naive derivation of the
4 2
modified IAM by choosing the subtraction points at the
where,forbrevity,wehavealreadyusedtheelasticunitar- Adler zeros.
ity condition Eq. (5). As usual, we will approximate Note, however, that in contrast to the derivation where
LC(cid:1)1=t(cid:2)’(cid:5)LC(cid:1)t =t2(cid:2),sincethatistheresultofthechiral we used an arbitrary z , now all subtractions have been
4 2 0
expansion at low energies where the integral is weighted. performed in the very low-energy region so that chiral
In the above relations, the pole contributionsPC(cid:1)1=t(cid:2)and expansions for pole terms are well justified within ChPT.
PC(cid:1)t =t2(cid:2) now come from a double and a triple pole, The price to pay is that in the right cut integral terms we
4 2
respectively, and read have approximated s by s , which is irrelevant in the
A 2
Adler zero region since the pole contributions dominate
1 t00(cid:1)s (cid:2)
PC(cid:1)1=t(cid:2)(cid:7) (cid:5) A ; (24) there, and a remarkably good approximation in the physi-
t0(cid:1)sA(cid:2)(cid:1)s(cid:5)sA(cid:2) 2t0(cid:1)sA(cid:2)2 cal and resonance regions. Of course, we have still used
exact elastic unitarity in the integrands over the physical
t (cid:1)s (cid:2) t0(cid:1)s (cid:2) cut, which ensures that the modified IAM satisfies exact
PC(cid:1)t =t2(cid:2)(cid:7) 4 2 (cid:4) 4 2
4 2 t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2)2 t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2) elastic unitarity.
2 2 2 2 2 2
t00(cid:1)s (cid:2)
(cid:4) 4 2 : (25) V. MODIFIEDIAM: UNEQUALMASSES.
2t0(cid:1)s (cid:2)2
2 2
When dealing with unequal masses, as in the (cid:3)K scat-
In addition, it is important to remark that these pole con-
teringcasethatwewilluseforreference,inadditiontothe
tributions diverge either on s or s , so that for s’s or
2 A 2 left and right cuts, there is also a circular cut centered at
s’sAtheyare,byfar,thedominantcontributions,sinceat q(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
thesametime therightandleft cuttermstendtozeroand s(cid:7)0 with radius m2K (cid:5)m2(cid:3) that contributes to the dis-
therefore become negligible. persion relation. This circular cut lies in the low-energy
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regionwithintheapplicabilityrangeofChPT,andthus,as nowget
we do with the left cut, we will approximate the inverse
t (cid:1)s(cid:2)2 (cid:4) t (cid:1)s (cid:2) (cid:1)s (cid:5)s (cid:2)
amplitude by its ChPT series to fourth order. Taking into AmIAM(cid:1)s(cid:2)(cid:7) 2 4 2(cid:4) (cid:5) 2(cid:4) A
account that t2(cid:1)s(cid:2) has no cuts, we will have CC(cid:1)1=t(cid:2)(cid:7) t02(cid:1)s2(cid:4)(cid:2)2 (cid:1)s(cid:5)s2(cid:4)(cid:2)2 (cid:1)s(cid:5)s2(cid:4)(cid:2)(cid:1)s(cid:5)sA(cid:2)
(cid:5)CC(cid:1)t =t2(cid:2), where CC stands for the circular cut contri- (cid:2) t (cid:1)s (cid:2)t00(cid:1)s (cid:2)(cid:3)(cid:5)
4 2 (cid:3) t0(cid:1)s (cid:2)(cid:5)t0(cid:1)s (cid:2)(cid:4) 4 2(cid:4) 2 2(cid:4) :
bution.Thisisthesameapproximationusedbeforeforthe 2 2(cid:4) 4 2(cid:4) t0(cid:1)s (cid:2)
left cut, and thereforewe obtain the same IAMdispersive 2 2(cid:4)
(31)
derivation. Hence, there is still the same problem with
Adler zeros and spuriouspoles. Then, Eq. (11) with AmIAM(cid:1)s(cid:2) above unitarizes the I (cid:7)
The solution givenin previoussections workssimilarly 1=2, J (cid:7)0 channel and has an Adler zero of the correct
wellfortheI (cid:7)3=2andJ (cid:7)0partialwave.However,for chiral order and has no spurious pole. Let us remark that
the I (cid:7)1=2, J (cid:7)0 (cid:3)K scattering channel, complications the above AmIAM(cid:1)s(cid:2) is a generalization also valid for the
ariseduetotheformoftheLOpartialwavet (cid:1)s(cid:2)whichhas equal mass case, since it is reduced to the AmIAM(cid:1)s(cid:2) we
2
two zeros instead of one. already obtained in Eq. (16) when t00(cid:1)s (cid:2)(cid:7)0.
2 2(cid:4)
Apartfromthisnaiveformalderivationwecanfollowa
5(cid:1)s(cid:5)s (cid:2)(cid:1)s(cid:5)s (cid:2)
t (cid:1)s(cid:2)(cid:7)(cid:5) 2(cid:4) 2(cid:5) ; (29) dispersive approach, detailed next, in which we make use
2 128(cid:3)fs
of analyticity and the ChPT series is only used within its
q(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) applicability region.
with s (cid:7)1(cid:1)m2 (cid:4)m2 (cid:11)2 4m4 (cid:5)7m2m2 (cid:4)4m4(cid:2) whose
2(cid:11) 5 K (cid:3) K K (cid:3) (cid:3)
values are s2(cid:4) (cid:7)0:24 GeV2, and s2(cid:5) (cid:7)(cid:5)0:13 GeV2. In B. Dispersive derivation at z , I(cid:7)1=2, J(cid:7)0 channel
0
particular, this means that, contrary to the previous cases,
ThederivationfollowsexactlythatofSec.IIA,butnow,
t00(cid:1)s (cid:2)(cid:1)0, which complicates the derivation of a modi-
2(cid:11) due to the additional zero, s , of t , which lies on the
fiedIAM.Nevertheless,onceagainwehavefound,forthis 2(cid:5) 2
negative axis, we cannot simply write, as usually done,
specialcaseoftheI (cid:7)1=2,J (cid:7)0channel,anaiveandtwo
R Disct =t2 (cid:7)2iR Imt =t2, nor R Disc1=t (cid:7)0,
dispersive derivations, that wedetail next. LC 4 2 LC 4 2 LC 2
where Discf (cid:7)f(cid:1)x(cid:4)i(cid:4)(cid:2)(cid:5)f(cid:1)x(cid:5)i(cid:4)(cid:2) with real x. This
zero also modifies the form of the dispersion relation for
A. Naive derivation, I(cid:7)1=2, J(cid:7)0 channel
1=t ,that nowreads
Following Sec. III, we define 1=tmIAM (cid:7)1=tIAM(cid:4) 2
AmIAM(cid:1)s(cid:2)=t2(cid:1)s(cid:2)withAmIAM(cid:1)s(cid:2)ananalyticfunction,atleast 1 1
outside the2real axis, and real for real s to preserve the t2(cid:1)s(cid:2)(cid:7)t2(cid:1)z0(cid:2)(cid:4)PC(cid:4)(cid:1)1=t2(cid:2)(cid:4)PC(cid:5)(cid:1)1=t2(cid:2);
unitarity andanalytic properties oftheoriginal amplitude. 1 (cid:2) 1 1 (cid:3) (32)
tNakexintgwienteoxpacacnodu1n=tttIhAaMtinnowLatu00r(cid:1)esn(cid:2)t(cid:1)se0ri,eosbatraoiunnindgs(cid:7)s2(cid:4), PC(cid:11)(cid:1)1=t2(cid:2)(cid:7)t02(cid:1)s2(cid:11)(cid:2) s(cid:5)s2(cid:11)(cid:5)z0(cid:5)s2(cid:11) :
2
Following the same steps as in Sec. IIA, we approximate
1 t (cid:1)s (cid:2) t0(cid:1)s (cid:2)(cid:5)t0(cid:1)s (cid:2)
(cid:7)(cid:5) 4 2 (cid:4) 2 2(cid:4) 4 2(cid:4) 1=t on the left cut by its chiral expansion, but now taking
tIAM(cid:1)s(cid:2) t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2)2 t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2) into account that, as mentioned above,
2 2(cid:4) 2(cid:4) 2 2(cid:4) 2(cid:4)
R
(cid:4) t4(cid:1)s2(cid:4)(cid:2)t020(cid:1)s2(cid:4)(cid:2) (cid:4)O(cid:1)s(cid:5)s (cid:2)0: (30) LCdzDisc(cid:1)1=t2(cid:1)z(cid:2)(cid:2)(cid:1)0, i.e.
t02(cid:1)s2(cid:4)(cid:2)3(cid:1)s(cid:5)s2(cid:4)(cid:2) 2(cid:4) LC(cid:1)1=t(cid:2)(cid:7)s(cid:5)z0 Z dz Disc1=t(cid:1)z(cid:2)
AsinSec.IIIwesubtractthepoleats andaddapoleat 2(cid:3)i LC (cid:1)z(cid:5)s(cid:2)(cid:1)z(cid:5)z0(cid:2)
2(cid:4)
sA totheinverseamplitude,andtheconstantinthesA pole ’s(cid:5)z0 Z dzDisc(cid:1)1=t2(cid:1)z(cid:2)(cid:5)t4(cid:1)z(cid:2)=t2(cid:1)z(cid:2)2(cid:2);
term is calculated perturbatively in order to match the
2(cid:3)i (cid:1)z(cid:5)s(cid:2)(cid:1)z(cid:5)z (cid:2)
LC 0
chiral expansion at low energies. Note that, following our
(33)
previous arguments, we do not need to subtract the s
2(cid:5)
pole, since Imt (cid:1)0 on the LC, so that no spurious pole where we now get (cid:5)LC(cid:1)t =t2(cid:2) as before, and a new term
4 4 2
willappearinthatregion.ProceedingthenasinSec.III,we coming from 1=t ,
2
s(cid:5)z0 Z dz Disc1=t2(cid:1)z(cid:2) (cid:7)s(cid:5)z0 Z dz(cid:1)1=t2(cid:1)z(cid:4)i(cid:4)(cid:2)(cid:5)1=t2(cid:1)z(cid:5)i(cid:4)(cid:2)(cid:2)
2(cid:3)i LC (cid:1)z(cid:5)s(cid:2)(cid:1)z(cid:5)z0(cid:2) 2(cid:3)i LC (cid:1)z(cid:5)s(cid:2)(cid:1)z(cid:5)z0(cid:2)
Z (cid:1)z(cid:5)s (cid:2)=t (cid:1)z(cid:2) 1 (cid:2) 1 1 (cid:3)
(cid:7)(cid:5)(cid:1)s(cid:5)z (cid:2) dz 2(cid:5) 2 (cid:8)(cid:1)z(cid:5)s (cid:2)(cid:7) (cid:5) (34)
0 (cid:1)z(cid:5)s(cid:2)(cid:1)z(cid:5)z (cid:2) 2(cid:5) t0(cid:1)s (cid:2) s(cid:5)s z (cid:5)s
LC 0 2 2(cid:5) 2(cid:5) 0 2(cid:5)
which is equal to PC (cid:1)1=t (cid:2), so they will cancel in the expression for 1=t. Thus we again obtain Eq. (7), but now the
(cid:5) 2
explicit expression forPC(cid:1)t =t2(cid:2) at s(cid:7)s is different from Eq. (18) because t00(cid:1)s(cid:2)(cid:1)0,
4 2 2(cid:4) 2
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t (cid:1)s (cid:2) (cid:2) 1 1 (cid:3) (cid:2)t0(cid:1)s (cid:2) t (cid:1)s (cid:2)t00(cid:1)s (cid:2)(cid:3)(cid:2) 1 1 (cid:3)
PC(cid:1)t =t2(cid:2)(cid:7) 4 2(cid:4) (cid:5) (cid:4) 4 2(cid:4) (cid:5) 4 2(cid:4) 2 2(cid:4) (cid:5) : (35)
4 2 t0(cid:1)s (cid:2)2 (cid:1)s(cid:5)s (cid:2)2 (cid:1)z (cid:5)s (cid:2)2 t0(cid:1)s (cid:2)2 t0(cid:1)s (cid:2)3 s(cid:5)s z (cid:5)s
2 2(cid:4) 2(cid:4) 0 2(cid:4) 2 2(cid:4) 2 2(cid:4) 2(cid:4) 0 2(cid:4)
Finally, weget latter is numerically small not only in the physical region
but also in the subthreshold region, so that it can be
t (cid:1)s(cid:2)2
tz0IAM(cid:1)s(cid:2)(cid:7) 2 ; (36) neglected toobtain again the result fromthe naive deriva-
t2(cid:1)s(cid:2)(cid:5)t4(cid:1)s(cid:2)(cid:4)Az0IAM(cid:1)s(cid:2) tion, thus justifying numerically the mIAMand the IAM.
with However,aderivationofthemIAMthatdiffersfromthe
standard one, since subtractions are made at the Adler
Az0IAM(cid:1)s(cid:2)(cid:7)t22(cid:1)s(cid:2)(cid:1)(cid:5)PC(cid:4)(cid:1)1=t2(cid:2)(cid:4)PC(cid:1)t4=t22(cid:2)(cid:4)PC(cid:1)1=t(cid:2)(cid:2); zeros, is also possible for the unequal mass case, and we
(37) detail it next.
where, again, we evaluate 1=t0(cid:1)s (cid:2) in PC(cid:1)1=t(cid:2) perturba-
A
tively,obtaining forA(cid:1)s(cid:2) C.DispersivederivationsubtractingattheAdlerzeros,
I(cid:7)1=2, J(cid:7)0 channel
t (cid:1)s(cid:2)2
Az0IAM(cid:1)s(cid:2)(cid:7)AmIAM(cid:1)s(cid:2)(cid:5)t2(cid:1)z (cid:2)2AmIAM(cid:1)z0(cid:2): (38) ThederivationfollowscloselythatinSec.IVB,butnow
2 0 we will have extra terms (not negligible in the chiral
Formally,thislooksthesameasEq.(21)butnowAmIAM(cid:1)s(cid:2) expansion) coming from Disc1=t in the left cut when s
2 A
is of a more general form. Indeed, we obtain again is expanded around s . In addition, the expressions for
2
AmIAM(cid:1)s(cid:2) plus a term depending on z , which is also the pole contributions are more complicated because
0
O(cid:1)f(cid:5)6(cid:2). Nevertheless we will see that, as long as z re- t00(cid:1)s(cid:2)(cid:1)0.
0 2
mains in the range of validity of the approximations, the We again expand the left cut to NLO:
s(cid:5)s (cid:5)s Z Disc(cid:1)1=t (cid:1)z(cid:2)(cid:5)t (cid:1)z(cid:2)=t (cid:1)z(cid:2)2(cid:2)(cid:2) s (cid:3)
LC(cid:1)1=t(cid:2)’ 2(cid:4) 4 dz 2 4 2 1(cid:4) 4
2(cid:3)i LC (cid:1)z(cid:5)s(cid:2)(cid:1)z(cid:5)s2(cid:4)(cid:2) z(cid:5)s2(cid:4)
s(cid:5)s t (cid:1)s (cid:2)
’(cid:5)LC(cid:1)t =t2(cid:2)(cid:4) 2(cid:4) (cid:5) 4 2(cid:4) ; (39)
4 2 t0(cid:1)s (cid:2)(cid:1)s(cid:5)s (cid:2)(cid:1)s (cid:5)s (cid:2) t0(cid:1)s (cid:2)t0(cid:1)s (cid:2)(cid:1)s (cid:5)s (cid:2)2
2(cid:5) 2(cid:5) 2(cid:5) 2(cid:4) 2 2(cid:4) 2 2(cid:5) 2(cid:4) 2(cid:5)
wherewe obtain (cid:5)LC(cid:1)t =t2(cid:2) plus two extra terms. The pole contributions nowread
4 2
1 t00(cid:1)s (cid:2)
PC(cid:1)1=t(cid:2)(cid:7) (cid:5) A ; (40)
t0(cid:1)s (cid:2)(cid:1)s(cid:5)s (cid:2) 2t0(cid:1)s (cid:2)2
A A A
t (cid:1)s (cid:2) (cid:2)t0(cid:1)s (cid:2) t (cid:1)s (cid:2)t00(cid:1)s (cid:2)(cid:3) 1 t00(cid:1)s (cid:2) t0(cid:1)s (cid:2)t00(cid:1)s (cid:2) 3t (cid:1)s (cid:2)t00(cid:1)s (cid:2)2
PC(cid:1)t =t2(cid:2)(cid:7) 4 2(cid:4) (cid:4) 4 2(cid:4) (cid:5) 4 2(cid:4) 2 2(cid:4) (cid:4) 4 2(cid:4) (cid:5) 4 2(cid:4) 2 2(cid:4) (cid:4) 4 2(cid:4) 2 2(cid:4)
4 2 t0(cid:1)s (cid:2)2(cid:1)s(cid:5)s (cid:2)2 t0(cid:1)s (cid:2)2 t0(cid:1)s (cid:2)3 s(cid:5)s 2t0(cid:1)s (cid:2)2 t0(cid:1)s (cid:2)3 4t0(cid:1)s (cid:2)4
2 2(cid:4) 2(cid:4) 2 2(cid:4) 2 2(cid:4) 2(cid:4) 2 2(cid:4) 2 2(cid:4) 2 2(cid:4)
t (cid:1)s (cid:2)t000(cid:1)s (cid:2)
(cid:5) 4 2(cid:4) 2 2(cid:4) ; (41)
3t0(cid:1)s (cid:2)3
2 2(cid:4)
wherealltermsinPC(cid:1)1=t(cid:2)willbeevaluatedperturbatively VI.NUMERICALRESULTS:COMPARISON
excepting 1=(cid:1)s(cid:5)s (cid:2), which is the one that gives the BETWEENDIFFERENTAPPROACHESANDz
A 0
needed pole at s to the inverse amplitude. Then, adding SENSITIVITY
A
the dispersion relations for1=t and t =t2, and using that
4 2 In this section we compare numerically the IAM with
themodifiedmethodswehavederived.Theprecisevalues
1 (cid:7) 1 (cid:4) s(cid:5)s2(cid:4) oftheChPTlow-energyconstantsarenotrelevantforwhat
t (cid:1)s(cid:2) t0(cid:1)s (cid:2)(cid:1)s(cid:5)s (cid:2) t0(cid:1)s (cid:2)(cid:1)s(cid:5)s (cid:2)(cid:1)s (cid:5)s (cid:2) we want to show here. Just for illustration, we take for
2 2 2(cid:4) 2(cid:4) 2 2(cid:5) 2(cid:5) 2(cid:5) 2(cid:4)
t00(cid:1)s (cid:2) SU(2) ChPT the typical values of lr3 (cid:7)0:82(cid:3)10(cid:5)3,
(cid:5) 2 2(cid:4) ; (42) lr (cid:7)6:2(cid:3)10(cid:5)3 from the second reference in [1], at a
2t0(cid:1)s (cid:2)2 4
2 2(cid:4) renormalization scale (cid:9)(cid:7)770 MeV. The values
lr (cid:7)(cid:5)3:7(cid:3)10(cid:5)3, lr (cid:7)5:0(cid:3)10(cid:5)3 have been obtained
1 2
wethusarriveatAmIAM inEq.(31)plusO(cid:1)f(cid:5)6(cid:2)termsthat from an IAM fit to (cid:3)(cid:3) scattering data. In particular, we
can be safely neglected since they all correspond to the have used the same data sets used in the IAM fits of the
chiral expansion at very lowenergies. second reference in [8], but we have updated the contro-
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GO´MEZ NICOLA,PELA´EZ, ANDR´IOS PHYSICALREVIEW D77, 056006 (2008)
versial f (cid:1)600(cid:2) channel by using the choice of data ex- z could be to s and s by looking at the expansion
0 0 A 2
plainedin[19]consistentwithforwarddispersionrelations
1 1 t (cid:1)z (cid:2) 1 (cid:2) s (cid:3)
andRoyequationsasshownin[20].For(cid:3)Kscatteringwe ’ (cid:5) 4 0 ’ 1(cid:4) 4 (cid:4)(cid:13)(cid:13)(cid:13) ;
have chosen the central values of the SU(cid:1)3(cid:2) ChPT con- t(cid:1)z0(cid:2) t2(cid:1)z0(cid:2) t2(cid:1)z0(cid:2)2 t2(cid:1)z0(cid:2) (cid:1)z0(cid:5)s2(cid:2)
stants of the IAMI set in the last reference of [8]. (43)
WeshowinFig.2severalplotscomparingtheIAMand
where we have only made explicit the s =(cid:1)z (cid:5)s (cid:2) pole
mIAMresultsforthemodulusofdifferentpartialwavesof 4 0 2
term. Hence, for our approximations to remain valid, we
(cid:3)(cid:3)and(cid:3)Kelasticscatteringinthescalarchannel.Wesee
have to make sure that the ratio s =(cid:1)z (cid:5)s (cid:2) is small
that both methods are indistinguishable in the physical 4 0 2
enough. Thus, we show in Fig. 3 the contour plots in the
region, shown in the left column, and only differ in the
energy and z plane ofthe relativedifference between the
Adler zero region which is shown in detail in the right 0
mIAMand the z IAM,
column. Note that the IAM has a spurious pole and a 0
double zero in that region, whereas the mIAM does not jtmIAM(cid:1)s(cid:2)(cid:5)tz0IAM(cid:1)s(cid:2)j
(cid:2)(cid:7) ; (44)
have sucha pole and its zero is simple. 1jtmIAM(cid:1)s(cid:2)(cid:4)tz0IAM(cid:1)s(cid:2)j
Wehavealsocalculatedthe(cid:5)and(cid:7)polepositionswith 2
the IAMand the mIAM,obtaining the same pole position as a function of the energy and z . We show two contour
0
within (cid:12)1 MeV, as shownin Table I. lines corresponding to (cid:2)(cid:7)10% and 5%. We see that, as
In the derivation of the z IAM, we have an arbitrary long as the choice of z is sufficiently far from s and s
0 0 A 2
subtraction point z (cid:1)s ;s , but we only know how to (thewhitelinesintheplots)theresultofthez IAMdiffers
0 2 A 0
calculate the amplitude at z if we can use the chiral little fromthemIAM.Inparticular,wehavecheckedthat,
0
expansion 1=t(cid:1)z (cid:2)’1=t (cid:1)z (cid:2)(cid:5)t (cid:1)z (cid:2)=t (cid:1)z (cid:2)2, which is in order to obtain a relative difference less than 10% and
0 2 0 4 0 2 0
only valid if z lies in the low-energy region. Also, due 5%,intheworstcase,whichistheI (cid:7)2,J (cid:7)0wave,we
0
tot havingazeroats ,theaboveexpansionisaverybad have to be sure that our choice of subtraction point z
2 2 0
approximation if z isnear s .We can estimate howclose makes the ratio of js =(cid:1)z (cid:5)s (cid:2)j<3% and 2%, respec-
0 2 4 0 2
I 2,J 0
I 0,J 0 0.08
0.075
5%
0.05 5% 0.06
10% 10%
0.025
2GeV 0 2GeV 0.04
z0 z0
0.025
0.02
0.05
0.075 0
0 200 400 600 800 1000 0 200 400 600 800 1000
s MeV s MeV
I 1 2,J 0 I 3 2,J 0
0.5 0.5
0.4 0.4
5%
10% 10%
5%
0.3 0.3
2V 2V
Ge Ge
z0 0.2 z0 0.2
0.1 0.1
0 0
400 500 600 700 800 900 1000 400 500 600 700 800 900 1000
s MeV s MeV
FIG. 3. ContourplotsoftherelativedifferencesbetweenthemIAMandthez IAMinthe(cid:1)s;z (cid:2)plane.Weseethat,aslongasz lies
0 0 0
inthelow-energyregionbutsufficientlyfarfromtheAdlerzeross ands (whitelines),thedifferencesbecomesmallforallpartial
A 2
waves. Note that for z (cid:7)s (black line) the differences for all cases are lessthan 5%.
0 th
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