We propose a direct test of the existence of gauge duals for
nonsupersymmetric asymptotically free gauge theories developing an infrared
fixed point by computing the S-parameter in the electric and dual magnetic
description. In particular we show that at the lower bound of the conformal
window the magnetic S-parameter, i.e. the one determined via the dual magnetic
gauge theory, assumes a simple expression in terms of the elementary magnetic
degrees of freedom. The results further support our recent conjecture of the
existence of a universal lower bound on the S parameter and indicates that it
is an ideal operator for counting the active physical degrees of freedom within
the conformal window. Our results can be directly used to unveil possible four
dimensional gauge duals and constitute the first explicit computation of a
nonperturbative quantity, in the electric variables, via nonsupersymmetric
gauge duality.Comment: RevTeX 4, 4 pages, two columns. Version to match the published
  version in Phys. Rev. Let

Sannino, Francesco

English

arXiv

We propose a direct test of the existence of gauge duals for nonsupersymmetric asymptotically free gauge theories developing an infrared fixed point by computing the S-parameter in the electric and dual magnetic description. In particular we show that at the lower bound of the conformal window the magnetic S-parameter, i.e. the one determined via the dual magnetic gauge theory, assumes a simple expression in terms of the elementary magnetic degrees of freedom. The results further support our recent conjecture of the existence of a universal lower bound on the S parameter and indicates that it is an ideal operator for counting the active physical degrees of freedom within the conformal window. Our results can be directly used to unveil possible four dimensional gauge duals and constitute the first explicit computation of a nonperturbative quantity, in the electric variables, via nonsupersymmetric gauge duality

Syddansk Universitets Forskerportal

       University of Southern DenmarkMagnetic S-parameterSannino, FrancescoPublished in:Physical Review LettersDOI:10.1103/PhysRevLett.105.232002Publication date:2010Citation for pulished version (APA):Sannino, F. (2010). Magnetic S-parameter. Physical Review Letters, 105, 232002.https://doi.org/10.1103/PhysRevLett.105.232002Go to publication entry in University of Southern Denmark's Research PortalTerms of useThis work is brought to you by the University of Southern Denmark.Unless otherwise specified it has been shared according to the terms for self-archiving.If no other license is stated, these terms apply:            • You may download this work for personal use only.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying this open access versionIf you believe that this document breaches copyright please contact us providing details and we will investigate your claim.Please direct all enquiries to puresupport@bib.sdu.dkDownload date: 22. Oct. 2025Magnetic S-parameterFrancesco Sanninor∗r CP3-Origins, Campusvej 55, DK-5230 Odense M, Denmark.†We propose a direct test of the existence of gauge duals for nonsupersymmetric asymptotically freegauge theories developing an infrared fixed point by computing the S-parameter in the electric anddual magnetic description. In particular we show that at the lower bound of the conformal windowthe magnetic S-parameter, i.e. the one determined via the dual magnetic gauge theory, assumesa simple expression in terms of the elementary magnetic degrees of freedom. The results furthersupport our recent conjecture of the existence of a universal lower bound on the S parameter andindicates that it is an ideal operator for counting the active physical degrees of freedom within theconformal window. Our results can be directly used to unveil possible four dimensional gauge dualsand constitute the first explicit computation of a nonperturbative quantity, in the electric variables,via nonsupersymmetric gauge duality.One of the most fascinating possibilities is that genericasymptotically free gauge theories have magnetic duals.In fact, in the late nineties, in a series of ground breakingpapers Seiberg [1] provided strong support for the exis-tence of a consistent picture of such a duality within a su-persymmetric framework. Using such a duality, Seiberghas been able to identify the boundary of the confor-mal window for supersymmetric QCD as function of thenumber of flavors and colors.Arguably the existence of a possible dual of a genericnonsupersymmetric asymptotically free gauge theoryable to reproduce its infrared dynamics must match the’t Hooft anomaly conditions [2]. We have exhibited sev-eral solutions of these conditions for QCD in [3] andfor certain gauge theories with higher dimensional rep-resentations in [4]. The novelty with respect to theseearlier results [5] are: i) The request that the gauge sin-glet operators associated to the magnetic baryons shouldbe interpreted as bound states of ordinary baryons [3];ii) The fact that the asymptotically free condition for thedual theory matches the lower bound on the conformalwindow obtained using the all orders beta function [6].These extra constraints help restricting further the num-ber of possible gauge duals without diminishing the ex-actness of the associate solutions with respect to the ’tHooft anomaly conditions.In this work we suggest a direct test of the possibleexistence of gauge duals using the VV − AA two-pointfunction determined in the conformal window of the un-derlying gauge theory upon the introduction of a massterm for the fermions and in the limit in which the ex-ternal momentum vanishes at a nonzero value of thefermion mass [7].We have shown in [7] that this parameter is exactly cal-culable, using the electric theory, near the upper limit ofthe conformal window. The reason is that there the elec-tric theory is in a perturbative regime. We argued that the†CP3- Origins: 2010-25results are important for the explorations of the confor-mal window since they shed light on relevant propertiesin this region. The results are also directly applicable tounparticle extensions of the standard model (SM) [8, 9].Near the lower bound of the conformal window we can-not compute analytically in a controllable way this pa-rameter but we expect the magnetic dual to be weaklycoupled and hence we can derive a closed form expres-sion for the S there via the gauge dual. We will refer toit as the magnetic S parameter (Sm).The oblique [10–13] parameter we use here is the onedefined in [14] and adapted for a theory developing aninfrared fixed point in [7]. As we have demonstrated in[7], within the conformal window, one must distinguishbetween two non-commuting limits once we introduce amass term for the fermions. To be precise we have shownthat if we take the fermion mass to zero, at finite externalmomentum, then the associated S-parameter vanishes,viceversa, if we send the external momentum to zerofirst the S-parameter never vanishes. We have also ar-gued that this is the limit which smoothly connects tothe S-parameter in the chirally broken phase relevant forbeyond SM applications. We will therefore concentrateon:limq2m2→0S . (1)The electric S-parameter (Se) is defined as the one com-puted using the electrical variables. Of course, if themagnetic and the electric theory are gauge duals of eachothers then Sm = Se. Near the electric (or magnetic)Banks-Zaks [15] infrared fixed point IRFP this parame-ter can be computed reliably by means of perturbationtheory [7] . We found that for an electric SU(N) gaugetheory with N f Dirac fermions transforming accordingto the representation r of the SU(N) gauge group, and asufficiently large number of flavors to be near the upperline of the conformal window, the leading terms in theq2/m2 expansion and at the leading perturbative order1arXiv:1007.0254v1  [hep-ph]  1 Jul 2010in the gauge coupling constant:limq2m2→0Se =]6π[1 +110x+170x2+ O(x−3)], (2)with x = m2q2 . The associated quantum global sym-metries of the underlying gauge theory are SUL(N f ) ×SUR(N f )×UV(1) if the fermion representation is complexor SU(2N f ) if real or pseudoreal. Here ] = ND d[r] countsthe number of doublets times the dimension of the repre-sentation d[r] under which the fermions transform. Forexample for the fundamental representation d[F] = N,for an SU(N) gauge group and d[S] = N(N + 1)/2 for thetwo-index symmetric representation of the gauge group.Note that given that we are in the conformal windowthe mass to the fermions is given via the standard Higgsmechanism.Consider the case of an underlying gauge group SU(3).The quantum flavor group of the massless theory is:SUL(N f ) × SUR(N f ) ×UV(1) . (3)The classical UA(1) symmetry is destroyed at the quan-tum level by the Adler-Bell-Jackiw anomaly. We indicatewith Qiα;c the two component left spinor where α = 1, 2is the spin index, c = 1, ..., 3 is the color index whilei = 1, ...,N f represents the flavor. Q̃α;ci is the two compo-nent conjugated right spinor. We summarize the trans-formation properties in the following table. The globalFields [SU(3)] SUL(N f ) SUR(N f ) UV(1)Q 1 1Q̃ 1 −1TABLE I: Fermion field content of an SU(3) gauge theory withquantum global symmetry SUL(N f ) × SUR(N f ) ×UV(1).anomalies are associated to the triangle diagrams fea-turing at the vertices three SU(N f ) generators (either allright or all left), or two SU(N f ) generators (all right or allleft) and one UV(1) charge. We indicate these anomaliesfor short with:SUL/R(N f )3 , SUL/R(N f )2 UV(1) . (4)For a vector like theory there are no further globalanomalies. The cubic anomaly factor, for fermions infundamental representations, is 1 for Q and −1 for Q̃while the quadratic anomaly factor is 1 for both leadingtoSUL/R(N f )3 ∝ ±3 , SUL/R(N f )2UV(1) ∝ ±3 . (5)We have computed the S-parameter in the pertur-bative regime of the conformal window, however wewould like now to determine this parameter near thelower bound of the conformal window. Here perturba-tion theory fails, in the electric variables, and one has toresort to other methods. However, if a magnetic gaugedual exists one expects it to be weakly coupled near thecritical number of flavors below which one breaks largedistance conformality in the electric variables. We canthen determine S near the lower boundary of the confor-mal window using perturbation theory in the magneticvariables. Determining a possible unique dual theoryfor QCD is, however, not simple given the few math-ematical constraints at our disposal. The saturation ofthe global anomalies is an important tool but is not ableto select out a unique solution. The goal is to find theexplicit expression for Sm in terms of the magnetic vari-ables by means of the most general expectation for thestructure of the gauge dual.As argued in [3–5] a candidate gauge dual theorywithin the conformal window, saturating the ’t Hooftanomaly conditions, would be constituted by an SU(X)gauge group with global symmetry group SUL(N f ) ×SUR(N f ) × UV(1) featuring magnetic quarks q and q̃ to-gether with SU(X) gauge singlet fermions identifiable asbaryons built out of the electric quarks Q. Since mesonsdo not affect directly global anomaly matching condi-tions we can add them to the spectrum of the dual theory.In particular they are needed to let the magnetic quarksand the gauge singlet fermions interact with each others.The new mesons will be massless and have no-self po-tential to respect the conformal invariance of the modelat large distances. We add to the magnetic quarks gaugesinglet Weyl fermions which can be identified with thebaryons of QCD but are, in fact, massless. The genericdual spectrum is summarized in table II. The wave func-tions for the gauge singlet fields A, C and S are obtainedby projecting the flavor indices of the following operatorεc1c2c3 Qi1c1 Qi2c2 Qi3c3 , (6)over the three irreducible representations of SUL(N f ) asindicated in the table II. These states are all singlets underthe SUR(N f ) flavor group. Similarly one can constructthe only right-transforming baryons Ã, C̃ and S̃ via Q̃.The B states are made by two Q fields and one rightfield Q̃ while the D fields are made by one Q and two Q̃fermions. y is the, yet to be determined, baryon chargeof the magnetic quarks while the baryon charge of com-posite states is fixed in units of the QCD quark one. The`s count the number of times the same baryonic matterrepresentation appears as part of the spectrum of the the-ory. Invariance under parity and charge conjugation ofthe underlying theory requires `J = ` J̃ with J = A,S, ...,Cand `B = −`D.The simplest mesonic operator is M ji and transformssimultaneously according to the antifundamental rep-resentation of SUL(N f ) and the fundamental represen-2Fields [SU(X)] SUL(N f ) SUR(N f ) UV(1) # of copiesq 1 y 1q̃ 1 −y 1A 1 1 3 `AS 1 1 3 `SC 1 1 3 `CBA 1 3 `BABS 1 3 `BSDA 1 3 `DADS 1 3 `DSÃ 1 1 −3 `ÃS̃ 1 1 −3 `S̃C̃ 1 1 −3 `C̃Mij 1 0 1TABLE II: Massless spectrum of magnetic quarks and baryonsand their transformation properties under the global symmetrygroup. The last column represents the multiplicity of each stateand each state is a Weyl fermion.tation of SUR(N f ). These states are not constrained byanomaly matching conditions and they mediate the in-teractions between the magnetic quarks and the gaugesinglet fermions via Yukawa-type interactions.To probe the chiral properties of the theory requiresadding a mass term for the fermions. Near the lower endof the conformal window the dual theory is expected tobe weakly coupled yielding the following expression forthe magnetic S-parameter:Sm = Sq + SB + SM , (7)withSq =ND6πX . (8)We will, however, consider here the case in which wegauge, with respect to the electroweak interactions, onlythe SUL(2) × SUR(2) subgroup where the hypercharge isthe diagonal generator of SU(2)R. In this case only onedoublet contributes directly to the S parameter, i.e, wecan set ND = 1. This parameter is still sensitive to thewhole dynamics. The spectrum of the magnetic quarks,baryons and mesons naturally splits into representationsof SUL(2)×SUL(N f−2)×SUR(2)×SUR(N f−2)×UV(1). Themagnetic quark q, with respect to this group, transformsaccording to:q→[( , 1, 1, 1)y ⊕ (1, , 1, 1)y]. (9)The baryons have the following decomposition underSUL(2) × SUL(N f − 2) × SUR(2) × SUR(N f − 2) ×UV(1):A →[(1, , 1, 1)3 ⊕ ( , , 1, 1)3 ⊕ (1, , 1, 1)3],S → [( , 1, 1, 1)3 ⊕ ( , , 1, 1)3 ⊕ ( , , 1, 1)3⊕, ⊕(1, , 1, 1)3]C → [( , 1, 1, 1)3 ⊕ (1, , 1, 1)3 ⊕ ( , , 1, 1)3⊕, ⊕( , , 1, 1)3 ⊕ (1, , 1, 1)3 ⊕ ( , , 1, 1)3]BA → [(1, 1, , 1)3 ⊕ (1, 1, 1, )3 ⊕ ( , , , 1)3⊕, ⊕( , , 1, )3 ⊕ (1, , , 1)3 ⊕ (1, , 1, )3]BS → [( , 1, , 1)3 ⊕ ( , 1, 1, )3 ⊕ ( , , , 1)3⊕, ⊕( , , 1, )3 ⊕ (1, , , 1)3 ⊕ (1, , 1, )3] .(10)The decomposition of the charged conjugated baryonsis obtained from the one above by exchanging left withright.Since we are gauging with respect to the elec-troweak theory the first two flavors we provide amass term to them as done in [16], i.e. via the in-troduction of a SM Higgs-type interaction. Since weare operating within the conformal window this isthe direct way to provide a mass to the fermions.By symmetry arguments we can pair only the stateswhich do not transform with respect to SUL(N f −2) × SUR(N f − 2) but still transform nontrivially un-der SUL(2) × SUR(2). These states are ( , 1, 1, 1)3 forthe baryon S; ( , 1, 1, 1)3 for C; (1, 1, , 1)3 for BA andfor BS the state ( , 1, , 1)3. We need to consider thecharge conjugated states as well. In terms of the spino-rial representations of SUL(2) ⊗ SUR(2) the states aboveare `S ( 32 , 0)3 ⊕ `C (12 , 0)3 ⊕ `BA (0,12 )3 ⊕ `BS (1,12 )3 with the` prefactor taking into account the multiplicity of eachstate. They will pair with their charged conjugatedfermion via the mass term operator of the typeψHψ̃withH the standard model Higgs field which transforms ac-cording to the ( 12 ,12 ) representation. Note that we canonly pair states with j2 = j1 ± 12 .Each pair of conjugated fermions transforming accord-ing to ( j1, j2)λ under SUL(2)×SUR(2)×UV(1) leads to thefollowing contribution to the Sm parameter [16]:Sb =2 db3π∑JJ′XJ,J′[2 f(m2J ,m2J′)+ g(m2J ,m2J′)]++ [ j−( j+ + 1)]29π ∑J2J + 1J(J + 1) , (11)with the index b indicating the specific baryon and db itsdegeneracy. We also have j− = | j1 − j2|, j+ = j1 + j2 andj− ≤ J ≤ j+ the total spin for each baryon contribution.If more than one spinorial representation belongs to thesame baryon b the contributions of all the states must3be taken into account. The nonvanishing components ofthe group theoretical factor XJ,J′ are:XJ,J =1 − ( j−( j+ + 1)J(J + 1))2 J(J + 1)(2J + 1)12 ,XJ,J−1 = XJ−1,J =−112(( j+ + 1)2 − J2) (J2 − j−2). (12)The functions f and g read [16]:f(m2J ,m2J′)= −6∫ 10dx x(1 − x) logxm2J + (1 − x)m2J′µ2 ,g(m2J ,m2J′)= 6∫ 10dxx(1 − x)mJmJ′xm2J + (1 − x)m2J′. (13)The mass of each fermion is directly proportional to theelectric fermion mass m and depends on the representa-tion according to the formula mJ = −mJ+ 12j1( j1+ 12 ). We havechosen as a reference energy scale µ = m. The contribu-tion of the baryon sector is then:SB =∑bSb . (14)The complex scalar meson M decomposes as:M→[( , 1, , 1) ⊕ ( , 1, 1, ) ⊕ (1, , , 1) ⊕ (1, , 1, 1, )].(15)Only the first state, ( 12 ,12 ), contributes to SM and leads to:SM =13π∑JJ′f(m2J ,m2J′). (16)with J, J′ = 1, 0, m2J = m20(1 + J(J + 1)). This is a differentmass parameterization than the one given in [16]. Wealso have m20 ∝ m2. All factors of order unity have beenset to unity and finally set the scaleµ = m0 in the functionf for the scalars. The contribution to SM vanishes unlessthere is a mass splitting between the different multipletsof the unbroken SU(2)V symmetry.Putting together the various terms we have for thenormalized Sm:6π3Sm =X3+`C + `BA3+25729`BS(32 log 2 − 39)− 0.14 .(17)The explicit dependence on the quark masses disappearfor the Sm parameter in agreement with the expectationfrom the leading contribution in q2/m2 to the Se param-eter. The above is the general expression for Sm nearthe lower end of the conformal window correspondingto the nonperturbative regime in the electric variables.From this expression is evident that the present defini-tion of the normalized S-parameter counts the relevantdegrees of freedom as function of the number of flavors.We estimate Sm using the possible dual provided in [3]for which X = 2N f − 15, `A = 2, `BA = −2 (we take +2since we are simply counting the states) with the other`s vanishing. Asymptotic freedom for the magnetic dualrequires at least N f = 9 for which 6πSm/3 = 1.523 whileif the lower bound of the conformal window occurs forN f = 10 we obtain 6πSm/3 = 2.19. Of course, only oneof these two values should be considered as the actualvalue of the normalized magnetic S parameter near thelower end of the electric conformal window. Both valuesare such that the normalized Sm is always larger than theelectrical one near the upper end of the conformal win-dow and are close to the one for two flavors QCD whichis around two [17].These results support our recent conjecture [7] accord-ing to which the normalized S parameter, obtained inthe limit when the external momentum vanishes at anonzero value of the quark mass, is a nondecreasingfunction of the number of flavors with respect to theunderlying electric theory satisfying a universal lowerbound corresponding to unity.∗ Electronic address: sannino@cp3.sdu.dk[1] N. Seiberg, Nucl. Phys. B 435, 129 (1995) [arXiv:hep-th/9411149].[2] G. ’t Hooft, Recent Developments in Gauge Theories, PlenumPress, 1980, 135; reprinted in Unity of Forces in the UniverseVol. II, A. Zee ed., World Scientific 1982, 1004.[3] F. Sannino, Phys. Rev. D 80, 065011 (2009) [arXiv:0907.1364[hep-th]].[4] F. Sannino, Nucl. Phys. B 830, 179 (2010) [arXiv:0909.4584[hep-th]].[5] J. Terning, Phys. Rev. Lett. 80, 2517 (1998) [arXiv:hep-th/9706074].[6] T. A. Ryttov and F. Sannino, Phys. Rev. D 78, 065001 (2008)[arXiv:0711.3745 [hep-th]].[7] F. Sannino, arXiv:1006.0207 [hep-lat].[8] H. Georgi, Phys. Rev. Lett. 98, 221601 (2007) [arXiv:hep-ph/0703260].[9] F. Sannino and R. Zwicky, Phys. Rev. D 79, 015016 (2009)[arXiv:0810.2686 [hep-ph]].[10] M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964(1990).[11] M. E. Peskin and T. Takeuchi, Phys. Rev. D 46, 381 (1992).[12] D. C. Kennedy and P. Langacker, Phys. Rev. Lett. 65, 2967(1990) [Erratum-ibid. 66, 395 (1991)].[13] G. Altarelli and R. Barbieri, Phys. Lett. B 253, 161 (1991).[14] H. J. He, N. Polonsky and S. f. Su, Phys. Rev. D 64, 053004(2001) [arXiv:hep-ph/0102144].[15] T. Banks and A. Zaks, Nucl. Phys. B 196, 189 (1982).[16] M. J. Dugan and L. Randall, Phys. Lett. B 264, 154 (1991).[17] P. A. Boyle, L. Del Debbio, J. Wennekers and J. M. Zanotti[RBC Collaborations and UKQCD Collaborations], Phys.Rev. D 81, 014504 (2010) [arXiv:0909.4931 [hep-lat]].4

Magnetic S-parameter

       University of Southern DenmarkMagnetic S-parameterSannino, FrancescoPublished in:Physical Review LettersDOI:10.1103/PhysRevLett.105.232002Publication date:2010Citation for pulished version (APA):Sannino, F. (2010). Magnetic S-parameter. Physical Review Letters, 105, 232002.https://doi.org/10.1103/PhysRevLett.105.232002Go to publication entry in University of Southern Denmark's Research PortalTerms of useThis work is brought to you by the University of Southern Denmark.Unless otherwise specified it has been shared according to the terms for self-archiving.If no other license is stated, these terms apply:            • You may download this work for personal use only.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying this open access versionIf you believe that this document breaches copyright please contact us providing details and we will investigate your claim.Please direct all enquiries to puresupport@bib.sdu.dkDownload date: 20. Oct. 2025Magnetic S-parameterFrancesco Sanninor∗r CP3-Origins, Campusvej 55, DK-5230 Odense M, Denmark.†We propose a direct test of the existence of gauge duals for nonsupersymmetric asymptotically freegauge theories developing an infrared fixed point by computing the S-parameter in the electric anddual magnetic description. In particular we show that at the lower bound of the conformal windowthe magnetic S-parameter, i.e. the one determined via the dual magnetic gauge theory, assumesa simple expression in terms of the elementary magnetic degrees of freedom. The results furthersupport our recent conjecture of the existence of a universal lower bound on the S parameter andindicates that it is an ideal operator for counting the active physical degrees of freedom within theconformal window. Our results can be directly used to unveil possible four dimensional gauge dualsand constitute the first explicit computation of a nonperturbative quantity, in the electric variables,via nonsupersymmetric gauge duality.One of the most fascinating possibilities is that genericasymptotically free gauge theories have magnetic duals.In fact, in the late nineties, in a series of ground breakingpapers Seiberg [1] provided strong support for the exis-tence of a consistent picture of such a duality within a su-persymmetric framework. Using such a duality, Seiberghas been able to identify the boundary of the confor-mal window for supersymmetric QCD as function of thenumber of flavors and colors.Arguably the existence of a possible dual of a genericnonsupersymmetric asymptotically free gauge theoryable to reproduce its infrared dynamics must match the’t Hooft anomaly conditions [2]. We have exhibited sev-eral solutions of these conditions for QCD in [3] andfor certain gauge theories with higher dimensional rep-resentations in [4]. The novelty with respect to theseearlier results [5] are: i) The request that the gauge sin-glet operators associated to the magnetic baryons shouldbe interpreted as bound states of ordinary baryons [3];ii) The fact that the asymptotically free condition for thedual theory matches the lower bound on the conformalwindow obtained using the all orders beta function [6].These extra constraints help restricting further the num-ber of possible gauge duals without diminishing the ex-actness of the associate solutions with respect to the ’tHooft anomaly conditions.In this work we suggest a direct test of the possibleexistence of gauge duals using the VV − AA two-pointfunction determined in the conformal window of the un-derlying gauge theory upon the introduction of a massterm for the fermions and in the limit in which the ex-ternal momentum vanishes at a nonzero value of thefermion mass [7].We have shown in [7] that this parameter is exactly cal-culable, using the electric theory, near the upper limit ofthe conformal window. The reason is that there the elec-tric theory is in a perturbative regime. We argued that the†CP3- Origins: 2010-25results are important for the explorations of the confor-mal window since they shed light on relevant propertiesin this region. The results are also directly applicable tounparticle extensions of the standard model (SM) [8, 9].Near the lower bound of the conformal window we can-not compute analytically in a controllable way this pa-rameter but we expect the magnetic dual to be weaklycoupled and hence we can derive a closed form expres-sion for the S there via the gauge dual. We will refer toit as the magnetic S parameter (Sm).The oblique [10–13] parameter we use here is the onedefined in [14] and adapted for a theory developing aninfrared fixed point in [7]. As we have demonstrated in[7], within the conformal window, one must distinguishbetween two non-commuting limits once we introduce amass term for the fermions. To be precise we have shownthat if we take the fermion mass to zero, at finite externalmomentum, then the associated S-parameter vanishes,viceversa, if we send the external momentum to zerofirst the S-parameter never vanishes. We have also ar-gued that this is the limit which smoothly connects tothe S-parameter in the chirally broken phase relevant forbeyond SM applications. We will therefore concentrateon:limq2m2→0S . (1)The electric S-parameter (Se) is defined as the one com-puted using the electrical variables. Of course, if themagnetic and the electric theory are gauge duals of eachothers then Sm = Se. Near the electric (or magnetic)Banks-Zaks [15] infrared fixed point IRFP this parame-ter can be computed reliably by means of perturbationtheory [7] . We found that for an electric SU(N) gaugetheory with N f Dirac fermions transforming accordingto the representation r of the SU(N) gauge group, and asufficiently large number of flavors to be near the upperline of the conformal window, the leading terms in theq2/m2 expansion and at the leading perturbative order1arXiv:1007.0254v1  [hep-ph]  1 Jul 2010in the gauge coupling constant:limq2m2→0Se =]6π[1 +110x+170x2+ O(x−3)], (2)with x = m2q2 . The associated quantum global sym-metries of the underlying gauge theory are SUL(N f ) ×SUR(N f )×UV(1) if the fermion representation is complexor SU(2N f ) if real or pseudoreal. Here ] = ND d[r] countsthe number of doublets times the dimension of the repre-sentation d[r] under which the fermions transform. Forexample for the fundamental representation d[F] = N,for an SU(N) gauge group and d[S] = N(N + 1)/2 for thetwo-index symmetric representation of the gauge group.Note that given that we are in the conformal windowthe mass to the fermions is given via the standard Higgsmechanism.Consider the case of an underlying gauge group SU(3).The quantum flavor group of the massless theory is:SUL(N f ) × SUR(N f ) ×UV(1) . (3)The classical UA(1) symmetry is destroyed at the quan-tum level by the Adler-Bell-Jackiw anomaly. We indicatewith Qiα;c the two component left spinor where α = 1, 2is the spin index, c = 1, ..., 3 is the color index whilei = 1, ...,N f represents the flavor. Q̃α;ci is the two compo-nent conjugated right spinor. We summarize the trans-formation properties in the following table. The globalFields [SU(3)] SUL(N f ) SUR(N f ) UV(1)Q 1 1Q̃ 1 −1TABLE I: Fermion field content of an SU(3) gauge theory withquantum global symmetry SUL(N f ) × SUR(N f ) ×UV(1).anomalies are associated to the triangle diagrams fea-turing at the vertices three SU(N f ) generators (either allright or all left), or two SU(N f ) generators (all right or allleft) and one UV(1) charge. We indicate these anomaliesfor short with:SUL/R(N f )3 , SUL/R(N f )2 UV(1) . (4)For a vector like theory there are no further globalanomalies. The cubic anomaly factor, for fermions infundamental representations, is 1 for Q and −1 for Q̃while the quadratic anomaly factor is 1 for both leadingtoSUL/R(N f )3 ∝ ±3 , SUL/R(N f )2UV(1) ∝ ±3 . (5)We have computed the S-parameter in the pertur-bative regime of the conformal window, however wewould like now to determine this parameter near thelower bound of the conformal window. Here perturba-tion theory fails, in the electric variables, and one has toresort to other methods. However, if a magnetic gaugedual exists one expects it to be weakly coupled near thecritical number of flavors below which one breaks largedistance conformality in the electric variables. We canthen determine S near the lower boundary of the confor-mal window using perturbation theory in the magneticvariables. Determining a possible unique dual theoryfor QCD is, however, not simple given the few math-ematical constraints at our disposal. The saturation ofthe global anomalies is an important tool but is not ableto select out a unique solution. The goal is to find theexplicit expression for Sm in terms of the magnetic vari-ables by means of the most general expectation for thestructure of the gauge dual.As argued in [3–5] a candidate gauge dual theorywithin the conformal window, saturating the ’t Hooftanomaly conditions, would be constituted by an SU(X)gauge group with global symmetry group SUL(N f ) ×SUR(N f ) × UV(1) featuring magnetic quarks q and q̃ to-gether with SU(X) gauge singlet fermions identifiable asbaryons built out of the electric quarks Q. Since mesonsdo not affect directly global anomaly matching condi-tions we can add them to the spectrum of the dual theory.In particular they are needed to let the magnetic quarksand the gauge singlet fermions interact with each others.The new mesons will be massless and have no-self po-tential to respect the conformal invariance of the modelat large distances. We add to the magnetic quarks gaugesinglet Weyl fermions which can be identified with thebaryons of QCD but are, in fact, massless. The genericdual spectrum is summarized in table II. The wave func-tions for the gauge singlet fields A, C and S are obtainedby projecting the flavor indices of the following operatorεc1c2c3 Qi1c1 Qi2c2 Qi3c3 , (6)over the three irreducible representations of SUL(N f ) asindicated in the table II. These states are all singlets underthe SUR(N f ) flavor group. Similarly one can constructthe only right-transforming baryons Ã, C̃ and S̃ via Q̃.The B states are made by two Q fields and one rightfield Q̃ while the D fields are made by one Q and two Q̃fermions. y is the, yet to be determined, baryon chargeof the magnetic quarks while the baryon charge of com-posite states is fixed in units of the QCD quark one. The`s count the number of times the same baryonic matterrepresentation appears as part of the spectrum of the the-ory. Invariance under parity and charge conjugation ofthe underlying theory requires `J = ` J̃ with J = A,S, ...,Cand `B = −`D.The simplest mesonic operator is M ji and transformssimultaneously according to the antifundamental rep-resentation of SUL(N f ) and the fundamental represen-2Fields [SU(X)] SUL(N f ) SUR(N f ) UV(1) # of copiesq 1 y 1q̃ 1 −y 1A 1 1 3 `AS 1 1 3 `SC 1 1 3 `CBA 1 3 `BABS 1 3 `BSDA 1 3 `DADS 1 3 `DSÃ 1 1 −3 `ÃS̃ 1 1 −3 `S̃C̃ 1 1 −3 `C̃Mij 1 0 1TABLE II: Massless spectrum of magnetic quarks and baryonsand their transformation properties under the global symmetrygroup. The last column represents the multiplicity of each stateand each state is a Weyl fermion.tation of SUR(N f ). These states are not constrained byanomaly matching conditions and they mediate the in-teractions between the magnetic quarks and the gaugesinglet fermions via Yukawa-type interactions.To probe the chiral properties of the theory requiresadding a mass term for the fermions. Near the lower endof the conformal window the dual theory is expected tobe weakly coupled yielding the following expression forthe magnetic S-parameter:Sm = Sq + SB + SM , (7)withSq =ND6πX . (8)We will, however, consider here the case in which wegauge, with respect to the electroweak interactions, onlythe SUL(2) × SUR(2) subgroup where the hypercharge isthe diagonal generator of SU(2)R. In this case only onedoublet contributes directly to the S parameter, i.e, wecan set ND = 1. This parameter is still sensitive to thewhole dynamics. The spectrum of the magnetic quarks,baryons and mesons naturally splits into representationsof SUL(2)×SUL(N f−2)×SUR(2)×SUR(N f−2)×UV(1). Themagnetic quark q, with respect to this group, transformsaccording to:q→[( , 1, 1, 1)y ⊕ (1, , 1, 1)y]. (9)The baryons have the following decomposition underSUL(2) × SUL(N f − 2) × SUR(2) × SUR(N f − 2) ×UV(1):A →[(1, , 1, 1)3 ⊕ ( , , 1, 1)3 ⊕ (1, , 1, 1)3],S → [( , 1, 1, 1)3 ⊕ ( , , 1, 1)3 ⊕ ( , , 1, 1)3⊕, ⊕(1, , 1, 1)3]C → [( , 1, 1, 1)3 ⊕ (1, , 1, 1)3 ⊕ ( , , 1, 1)3⊕, ⊕( , , 1, 1)3 ⊕ (1, , 1, 1)3 ⊕ ( , , 1, 1)3]BA → [(1, 1, , 1)3 ⊕ (1, 1, 1, )3 ⊕ ( , , , 1)3⊕, ⊕( , , 1, )3 ⊕ (1, , , 1)3 ⊕ (1, , 1, )3]BS → [( , 1, , 1)3 ⊕ ( , 1, 1, )3 ⊕ ( , , , 1)3⊕, ⊕( , , 1, )3 ⊕ (1, , , 1)3 ⊕ (1, , 1, )3] .(10)The decomposition of the charged conjugated baryonsis obtained from the one above by exchanging left withright.Since we are gauging with respect to the elec-troweak theory the first two flavors we provide amass term to them as done in [16], i.e. via the in-troduction of a SM Higgs-type interaction. Since weare operating within the conformal window this isthe direct way to provide a mass to the fermions.By symmetry arguments we can pair only the stateswhich do not transform with respect to SUL(N f −2) × SUR(N f − 2) but still transform nontrivially un-der SUL(2) × SUR(2). These states are ( , 1, 1, 1)3 forthe baryon S; ( , 1, 1, 1)3 for C; (1, 1, , 1)3 for BA andfor BS the state ( , 1, , 1)3. We need to consider thecharge conjugated states as well. In terms of the spino-rial representations of SUL(2) ⊗ SUR(2) the states aboveare `S ( 32 , 0)3 ⊕ `C (12 , 0)3 ⊕ `BA (0,12 )3 ⊕ `BS (1,12 )3 with the` prefactor taking into account the multiplicity of eachstate. They will pair with their charged conjugatedfermion via the mass term operator of the typeψHψ̃withH the standard model Higgs field which transforms ac-cording to the ( 12 ,12 ) representation. Note that we canonly pair states with j2 = j1 ± 12 .Each pair of conjugated fermions transforming accord-ing to ( j1, j2)λ under SUL(2)×SUR(2)×UV(1) leads to thefollowing contribution to the Sm parameter [16]:Sb =2 db3π∑JJ′XJ,J′[2 f(m2J ,m2J′)+ g(m2J ,m2J′)]++ [ j−( j+ + 1)]29π ∑J2J + 1J(J + 1) , (11)with the index b indicating the specific baryon and db itsdegeneracy. We also have j− = | j1 − j2|, j+ = j1 + j2 andj− ≤ J ≤ j+ the total spin for each baryon contribution.If more than one spinorial representation belongs to thesame baryon b the contributions of all the states must3be taken into account. The nonvanishing components ofthe group theoretical factor XJ,J′ are:XJ,J =1 − ( j−( j+ + 1)J(J + 1))2 J(J + 1)(2J + 1)12 ,XJ,J−1 = XJ−1,J =−112(( j+ + 1)2 − J2) (J2 − j−2). (12)The functions f and g read [16]:f(m2J ,m2J′)= −6∫ 10dx x(1 − x) logxm2J + (1 − x)m2J′µ2 ,g(m2J ,m2J′)= 6∫ 10dxx(1 − x)mJmJ′xm2J + (1 − x)m2J′. (13)The mass of each fermion is directly proportional to theelectric fermion mass m and depends on the representa-tion according to the formula mJ = −mJ+ 12j1( j1+ 12 ). We havechosen as a reference energy scale µ = m. The contribu-tion of the baryon sector is then:SB =∑bSb . (14)The complex scalar meson M decomposes as:M→[( , 1, , 1) ⊕ ( , 1, 1, ) ⊕ (1, , , 1) ⊕ (1, , 1, 1, )].(15)Only the first state, ( 12 ,12 ), contributes to SM and leads to:SM =13π∑JJ′f(m2J ,m2J′). (16)with J, J′ = 1, 0, m2J = m20(1 + J(J + 1)). This is a differentmass parameterization than the one given in [16]. Wealso have m20 ∝ m2. All factors of order unity have beenset to unity and finally set the scaleµ = m0 in the functionf for the scalars. The contribution to SM vanishes unlessthere is a mass splitting between the different multipletsof the unbroken SU(2)V symmetry.Putting together the various terms we have for thenormalized Sm:6π3Sm =X3+`C + `BA3+25729`BS(32 log 2 − 39)− 0.14 .(17)The explicit dependence on the quark masses disappearfor the Sm parameter in agreement with the expectationfrom the leading contribution in q2/m2 to the Se param-eter. The above is the general expression for Sm nearthe lower end of the conformal window correspondingto the nonperturbative regime in the electric variables.From this expression is evident that the present defini-tion of the normalized S-parameter counts the relevantdegrees of freedom as function of the number of flavors.We estimate Sm using the possible dual provided in [3]for which X = 2N f − 15, `A = 2, `BA = −2 (we take +2since we are simply counting the states) with the other`s vanishing. Asymptotic freedom for the magnetic dualrequires at least N f = 9 for which 6πSm/3 = 1.523 whileif the lower bound of the conformal window occurs forN f = 10 we obtain 6πSm/3 = 2.19. Of course, only oneof these two values should be considered as the actualvalue of the normalized magnetic S parameter near thelower end of the electric conformal window. Both valuesare such that the normalized Sm is always larger than theelectrical one near the upper end of the conformal win-dow and are close to the one for two flavors QCD whichis around two [17].These results support our recent conjecture [7] accord-ing to which the normalized S parameter, obtained inthe limit when the external momentum vanishes at anonzero value of the quark mass, is a nondecreasingfunction of the number of flavors with respect to theunderlying electric theory satisfying a universal lowerbound corresponding to unity.∗ Electronic address: sannino@cp3.sdu.dk[1] N. Seiberg, Nucl. Phys. 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B 253, 161 (1991).[14] H. J. He, N. Polonsky and S. f. Su, Phys. Rev. D 64, 053004(2001) [arXiv:hep-ph/0102144].[15] T. Banks and A. Zaks, Nucl. Phys. B 196, 189 (1982).[16] M. J. Dugan and L. Randall, Phys. Lett. B 264, 154 (1991).[17] P. A. Boyle, L. Del Debbio, J. Wennekers and J. M. Zanotti[RBC Collaborations and UKQCD Collaborations], Phys.Rev. D 81, 014504 (2010) [arXiv:0909.4931 [hep-lat]].4

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Magnetic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math>Parameter

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Magnetic S-parameter

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