In Nucl. Phys. B391 (1993) 127, the value of the oblique correction parameter
S for walking technicolor theories was estimated using a technique called
Analytic Continuation by Duality (ACD). We apply the ACD technique to the
perturbative vacuum polarization function and find that it fails to reproduce
the well known result S=1/6\pi. This brings into question the reliability of
the ACD technique and the ACD estimate of S.Comment: 8 pages, LaTeX, 1 postscript figure. Uses cite.sty, sprocl.sty, and
  epsfig.sty. Talk presented at the 1996 International Workshop on Perspectives
  of Strong Coupling Gauge Theories (SCGT'96), 13-16 Nov. 1996, Nagoy

Goonetileke, L. C.

Ignjatovic, S. R.

Takeuchi, T.

Wijewardhana, L. C. R.

English

arXiv

In Nucl. Phys. B391 (1993) 127, Sundrum and Hsu estimated the value of the oblique correction parameter S for walking technicolor theories using a technique called Analytic Continuation by Duality (ACD). We apply the ACD technique to the perturbative vacuum polarization function and find that it fails to reproduce the well known result S=1/6\pi. This brings into question the reliability of the ACD technique and the Sundrum and Hsu estimate of S.In Nucl. Phys. B391 (1993) 127, Sundrum and Hsu estimated the value of the oblique correction parameter S for walking technicolor theories using a technique called Analytic Continuation by Duality (ACD). We apply the ACD technique to the perturbative vacuum polarization function and find that it fails to reproduce the well known result S=1/6\pi. This brings into question the reliability of the ACD technique and the Sundrum and Hsu estimate of S

Goonetileke, L.C.

Ignjatovic, S.R.

Wijewardhana, L.C.R.

CERN Document Server

UCTP{2/97VPI{IPPAP{97{2CERN{TH/96{198hep{ph/9702439AN EVALUATION OFTHE ANALYTIC CONTINUATION BY DUALITYTECHNIQUET. TakeuchiInstitute for Particle Physics and Astrophysics,Physics Department, Virginia Tech, Blacksburg, VA 24061-0435 aL. C. Goonetileke, S. R. Ignjatovic, and L. C. R. WijewardhanaDepartment of Physics, University of Cincinnati, Cincinnati, OH 45221AbstractIn Ref. 1, Sundrum and Hsu estimated the value of the oblique cor-rection parameter S for walking technicolor theories using a techniquecalled Analytic Continuation by Duality (ACD). We apply the ACD tech-nique to the perturbative vacuum polarization function and nd that itfails to reproduce the well known result S = 1=6. This brings intoquestion the reliability of the ACD technique and the Sundrum and Hsuestimate of S.Talk presented at the 1996 International Workshop onPerspectives of Strong Coupling Gauge Theories (SCGT’96)13{16 November, 1996, Nagoya, Japan.UCTP{2/96VPI{IPPAP{97{2CERN{TH/96{198hep{ph/9702439February 1997aCurrent address.AN EVALUATION OFTHE ANALYTIC CONTINUATION BY DUALITY TECHNIQUET. TAKEUCHI bTH Division, CERN, CH{1211 Geneve 23, SwitzerlandL. C. GOONETILEKE, S. R. IGNJATOVIC, and L. C. R. WIJEWARDHANADepartment of Physics, University of Cincinnati, Cincinnati, OH 45221In Ref. 1, Sundrum and Hsu estimated the value of the oblique correction parameterS for walking technicolor theories using a technique called Analytic Continuationby Duality (ACD). We apply the ACD technique to the perturbative vacuum polar-ization function and nd that it fails to reproduce the well known result S = 1=6.This brings into question the reliability of the ACD technique and the Sundrumand Hsu estimate of S.1 IntroductionThe analytic continuation by duality (ACD) technique was proposed by Sun-drum and Hsu in Ref. 1 as a reliable method to compute the oblique correctionparameter S for technicolor theories. The advertised advantage of the ACDtechnique was that it could be applied to both QCD{like and walking techni-color 2 theories whereas the dispersion relation technique used by Peskin andone of us in Ref. 3 could only be applied to the former. Furthermore, the ACDestimate of S for walking technicolor implied that walking dynamics couldrender S negative, making it compatible with the current experimental limit.4 This was in contrast to the result of Harada and Yoshida 5 who used theBethe{Salpeter equation approach to conclude that S was positive even forwalking theories.In this talk, we investigate the reliability of the ACD technique to seewhether the Sundrum{Hsu estimate should be taken seriously or not. In sec-tion 2, we review the denition of the S parameter and explain the ACDtechnique. In section 3, we apply the ACD technique to the perturbative spec-tral function to see if the famous result 1=6 could be reproduced. Discussionsand conclusions are stated in Section 4.2 The ACD techniqueThe S parameter, as dened in Ref. 3, is equal to a certain linear combina-tion of electroweak vacuum polarization functions evaluated at zero momentumbPresenting author.1Figure 1: The contour C which avoids the branch cut along the real s{axis.transfer. We represent this schematically asS = (0):(The precise denition of (s) is irrelevant to our ensuing discussion.) Thevacuum polarization function (s) is analytic in the entire complex s planeexcept for a branch cut along the positive real s axis starting from the lowestparticle threshold contributing to (s). Applying Cauchy’s theorem to thecontour C shown in Fig. 1, we ndS =1Z Rs0dsIm(s)s+12iIjsj=Rds(s)s: (1)If the radius of the contour R is taken to innity, the integral around the circleat jsj = R can be shown to vanish and we obtain the dispersion relationS =1Z Rs0dsIm(s)s;which was used in Ref. 3 to calculate S. However, the dispersion relationapproach requires the knowledge of Im(s) along the real s axis which is onlyavailable for QCD{like technicolor theories.The basic idea of the ACD technique, on the other hand, is to approximatethe kernel 1=s by a polynomial1s pN (s) =NXn=0an(N)sn; s 2 [s0; R]2and use it to make the integral along the real s axis vanish instead. ApplyingCauchy’s theorem to the product pN (s)(s) over the same contour C yields0 =1Z Rs0ds pN (s)Im(s) +12iIjsj=Rds pN (s)(s): (2)Subtracting Eq. (2) from Eq. (1), we obtainS = SN + t;whereSN 12iIjsj=Rds1s− pN (s)(s);t 1Z Rs0ds1s− pN (s)Im(s):For suciently large N , we can expect t to be negligibly small. In fact, itconverges to zero in the limit N ! 1 (though how quickly the convergenceoccurs depends on the interval [s0; R]). We can therefore neglect it and ap-proximate S with SN which is an integral around the circle jsj = R only. Wecall t the t error.If the radius of the contour R is taken to be suciently large, the function(s) can be approximated on jsj = R by a large momentum expansion:(s) MXm=1bm(s)sm: (3)This expression is obtained by analytically continuing the operator productexpansion (OPE) of (s) from the deep Euclidean region where it can becalculated for both QCD{like and walking technicolor theories. Therefore, wecan writeSN = SN;M + tr;whereSN;M 12iIjsj=Rds1s− pN (s) MXm=1bm(s)sm;tr 12iIjsj=Rds1s− pN (s) "(s)−MXm=1bm(s)sm#;3and approximate SN with SN;M . The neglected term tr is called the trunca-tion error.Sundrum and Hsu take their approximation one step further and neglectthe s{dependence of expansion coecients in Eq. (3), i.e.bm(s)  bm(−R)  b^m:This is obviously a dangerous approximation to make since the analytic struc-ture of the integrand will be completely altered. DeneSN;M = SACD + ACwhereSACD 12iIjsj=Rds1s− pN(s) MXm=1b^msm;AC 12iIjsj=Rds1s− pN(s) MXm=1bm(s)− b^msM:Sundrum and Hsu argue that AC can be expected to be highly suppressedand thus negligible since the dierence 1=s − pN (s) is approximately zero inthe vicinity of the positive real s axis where the dierence bm(s)− b^m can beexpected to be most pronounced. Thus:S  SACD:In this approximation, the integral for SACD will only pick up the residues ofthe single poles inside the integration contour and we nd,SACD = −minfN;M−1gXn=0an(N)b^n+1:We will call AC the analytical continuation error.To summarize, the ACD technique uses the relationS = SACD + AC + tr + t;and assumes that all three types of error can be neglected and approximatesS with SACD.4Table 1: SACD and the t, truncation, and analytical continuation errors for the perturbativevacuum polarization function. The cutos are [s0; R] = [4m2; 25m2], and the t routine wasthe least square t. The exact value of S is 1=6 = 0:0531.N M SACD SN;M = SACD + AC t tr3 2 0:2930 0:0580 −0:0002 −0:00483 0:2883 0:0530 0:00024 0:2884 0:0532 −0:00004 2 0:4330 0:0632 −0:0001 −0:01013 0:4203 0:0521 −0:00104 0:4211 0:0532 −0:00005 0:4211 0:0531 0:00005 3 0:5731 0:0506 −0:0000 0:00254 0:5759 0:0533 −0:00025 0:5757 0:0531 0:00006 0:5757 0:0531 −0:00003 The Perturbative Spectral FunctionTo check validity of the approximation S  SACD, we calculate SACD forthe one{loop contribution of a massive fermion doublet to S. The vacuumpolarization function (s) in this case is given by:pert(s) = −1m2sZ 10dx logh1− x(1− x)sm2i: (4)Evaluating this expression at s = 0, we nd the well known result S = 1=6.The function pert(s) is analytic in the entire complex s plane except for abranch cut along the positive real s axis starting from s = 4m2. The imaginarypart of this function along the cut is given byImpert(s) =m2s(s − 4m2);  =r1−4m2s: (5)The rst few terms of the large s-expansion of pert(s) are given bypert(s)= x− ln−1x+ 2+ x22 ln−1x+ 2+ x32 ln−1x− 1+ : : : ;where x  4m2=s. Using these expressions, we calculated SACD, AC, tr,and t. The results of our calculations for several values of N and M are5shown in Table 3. The t interval was [s0; R] = [4m2; 25m2], and the t routinewas the least square t.As is evident from Table. 3, the t and truncation errors are under excellentcontrol and SN;M reproduces the exact value of S accurately already at N =M = 3. However, the analytic continuation error is not. For the N = 5 case,for instance, SACD is larger than the exact value by more than an order ofmagnitude. In fact, we nd that SACD and AC diverge as N !1.We conclude that neglecting the s{dependence of the bm(s)’s fails mis-erably as an approximation. The reason for this can be traced to the factthat even though the dierence 1=s − pN (s) converges to zero within its ra-dius of convergence, outside it diverges. Therefore, the handwaving argumentof Sundrum and Hsu was wrong: the error induced by the neglect of the s{dependence of the bm(s)’s may be suppressed near the real s axis, but it isactually enhanced away from it.4 Discussion and ConclusionsThe application of the ACD technique to the perturbative vacuum polarizationfunction has shown that the analytic continuation error AC is not undercontrol and that the approximation S  SACD cannot be trusted. This bringsinto doubt the reliability of the ACD estimate of S obtained by Sundrum andHsu in Ref. 1.A natural question to ask next is whether the ACD technique can beimproved by including the s{dependence of the large momentum expansioncoecients bm(s) and using SN;M as the estimate of S instead of SACD. Inthe perturbative case, we have seen that this is an excellent approximation.However, whether SN;M will reproduce the correct value of S for all cases isfar from clear. If the large momentum expansion is an asymptotic series, thetruncation error tr may not converge to zero in the limit M ! 1. Even ifit is a convergent series, the convergence may be too slow for the method tobe practical. These questions, and all related problems will be addressed insubsequent papers. 6AcknowledgmentsWe would like to thank M. E. Peskin and K. Takeuchi for helpful discus-sions. This work was supported in part by the United States Departmentof Energy under Contract Number DE{AC02{76CH03000 (T.T.) and GrantNumber DE{FG02{84ER40153 (L.C.G., S.R.I., and L.C.R.W.).6References1. R. Sundrum and S. D. H. Hsu, Nucl. Phys. B391 127 (1993).2. B. Holdom, Phys. Lett. B150 301 (1985)T. Appelquist, D. Karabali, and L. C. R. Wijewardhana,Phys. Rev. Lett. 57 957 (1986);T. Appelquist and L. C. R. Wijewardhana,Phys. Rev. D35 774 (1987);Phys. Rev. D36 568 (1987).3. M. E. Peskin and T. Takeuchi,Phys. Rev. Lett. 65 964 (1990); Phys. Rev. D46 381 (1992);T. Takeuchi, in the Proceedings of the International Workshop on Elec-troweak Symmetry Breaking, Hiroshima, Japan, November 1991, editedby W. A. Bardeen, J. Kodaira, and T. Muta (World Scientic, Singapore,1992).4. J. L. Hewett, T. Takeuchi, and S. Thomas,SLAC{PUB{7088, CERN{TH/96{56, hep{ph/9603391 (March 1996).5. M. Harada and Y. Yoshida, Phys. Rev. D50 6902 (1994).6. S. R. Ignjatovic, T. Takeuchi, and L. C. R. Wijewardhana,Report No. UCTP{3/97, VPI{IPPAP{97{3, hep{ph/9702440,L. C. Goonetileke, S. R. Ignjatovic, T. Takeuchi, andL. C. R. Wijewardhana (in preparation).7

An evaluation of the analytic continuation by duality technique

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