We investigate the chiral dynamics of gauge theories developing an infrared
stable fixed point. We determine the dependence of the bilinear fermion
condensate on the underlying fermion mass and its anomalous dimension. We
introduce the instanton contributions and investigate how they affect the
dynamics near the fixed point. We generalize the Gell-Mann Oakes Renner
relation and suggest to use it to uncover the presence of an infrared fixed
point of the underlying gauge theory. Our results have an immediate impact on
the construction of sensible extensions of the Standard Model of particle
interactions and the general understanding of the phase diagram of strongly
coupled theories.Comment: 4 RevTex pages, 1 figure with small modifications and added
  reference

A. Hietanen

Francesco Sannino

English

arXiv

Crossref

Conformal chiral dynamics

We investigate the chiral dynamics of gauge theories developing an infrared stable fixed point. We determine the dependence of the bilinear fermion condensate on the underlying fermion mass and its anomalous dimension. We introduce the instanton contributions and investigate how they affect the dynamics near the fixed point. We generalize the Gell-Mann Oakes Renner relation and suggest to use it to uncover the presence of an infrared fixed point of the underlying gauge theory. Our results have an immediate impact on the construction of sensible extensions of the Standard Model of particle interactions and the general understanding of the phase diagram of strongly coupled theories

Sannino, Francesco

Syddansk Universitets Forskerportal

       University of Southern DenmarkConformal Chiral DynamicsSannino, FrancescoPublished in:Physical Review DDOI:10.1103/PhysRevD.80.017901Publication date:2009Citation for pulished version (APA):Sannino, F. (2009). Conformal Chiral Dynamics. Physical Review D, D80, 017901.https://doi.org/10.1103/PhysRevD.80.017901Go 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. 2025Conformal Chiral DynamicsFrancesco Sannino∗High Energy Center, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark.We investigate the chiral dynamics of gauge theories developing an infrared stable fixed point. Wedetermine the dependence of the bilinear fermion condensate on the underlying fermion mass andits anomalous dimension. We introduce the instanton contributions and investigate how they affectthe dynamics near the fixed point. We generalize the Gell-Mann Oakes Renner relation and suggestto use it to uncover the presence of an infrared fixed point of the underlying gauge theory. Ourresults have an immediate impact on the construction of sensible extensions of the Standard Modelof particle interactions and the general understanding of the phase diagram of strongly coupledtheories.Non-abelian gauge theories exist in a number of dis-tinct phases which can be classified according to the char-acteristic dependence of the potential energy on the dis-tance between two well separated static sources. Thecollection of all of these different behaviors, when repre-sented, for example, in the flavor-color space, constitutesthe Phase Diagram of the given gauge theory.Knowing the phase diagram of strongly coupled theo-ries has an immediate impact on the construction of sen-sible extensions of the Standard Model (SM) of particleinteractions [1]. Dynamical breaking of the electroweaksymmetry is a time-honored example [2, 3]. The use offermions transforming according to higher dimensionalrepresentations of the new gauge group is leading to sev-eral interesting phenomenological possibilities [4, 5, 6]such as Minimal Walking Technicolor (MWT)[7] and Ul-tra Minimal Walking Technicolor (UMT) [8]. These mod-els lie close, in theory space, to theories with non-trivialinfrared fixed points (IRFP)s [4, 9]. In the vicinity ofsuch a zero of the beta-function the coupling constantflows slowly, i.e. walks [10, 11, 12, 13]. Knowledge of thephase diagram is relevant also to provide natural ultra-violet completions of unparticle [14] models [1, 15]. Togain analytic insight one can use the conjectured all-orderbeta function for nonsupersymmetric gauge theories [9]together with the constraints from the unitarity of theconformal operators [16]. Other approaches are based onthe older truncated Schwinger-Dyson approach (SD) [17]or more recent ideas [18]. The analytical phase diagramobtained by this approach, and a comparison of it to re-cent lattice results [19, 20, 21, 22, 23, 24, 25, 26, 27], issummarized in [1].The goal here is to study the chiral properties at theIRFP, i.e. the conformal chiral dynamics.Conformal Chiral Dynamics: Our starting point is anonsupersymmetric non-abelian gauge theory with suf-ficient massless fermionic matter to develop a nontriv-ial IRFP. The cartoon of the running of the couplingconstant is represented in Fig. 1. In the plot ΛU isthe dynamical scale below which the IRFP is essentiallyreached. It can be defined as the scale for which α is2/3 of the fixed point value in a given renormalizationscheme. If the theory possesses an IRFP the chiral con-�Energy�U�*FIG. 1: Running of the coupling constant in an asymptoti-cally free gauge theory developing an infrared fixed point fora value α = α∗.densate must vanish at large distances. Here we want tostudy the behavior of the condensate when a flavor sin-glet mass term is added to the underlying Lagrangian:∆L = −mψ̃ψ + h.c. , (1)with m the fermion mass and ψfc as well as ψ̃cf left trans-forming two component spinors, c and f represent colorand flavor indices. The omitted color and flavor indices,in the Lagrangian term, are contracted. We consider firstthe case of fermionic matter in the fundamental represen-tation of the SU(N) gauge group. We then generalize ourresults to the case of higher dimensional representations.The effect of such a term is to break the conformalsymmetry together with some of the global symmetriesof the underlying gauge theory. The composite operator:O eψψf ′f = ψ̃f ′ψf , (2)has mass dimension d eψψ = 3− γm with γm the anoma-lous dimension of the mass term. At the fixed point γmis a positive number smaller than two [16]. We assumem ΛU . Dimensional analysis demands:∆L→ −mΛγmU Tr[O eψψ] + h.c. . (3)The mass term is a relevant perturbation around theIRFP driving the theory away from the fixed point.arXiv:0811.0616v3  [hep-ph]  2 Feb 20092It will induce a nonzero vacuum expectation value forO eψψ itself proportional to δf ′f . It is convenient to defineTr[O eψψ] = NfO with O a flavor singlet operator. Therelevant low energy Lagrangian term is then:−mΛγmU NfO + h.c. . (4)To determine the vacuum expectation value of O we re-place it, formally, with a sum over an infinite number ofcanonically normalized single particle states [28]:O(x) =∞∑n=1fnϕn(x) . (5)Each state possesses a mass Mn whose value is controlledby an artificial mass gap ∆ and a function of n, call itz(n), with the properties z(n+ 1) > z(n) and z(1) = 1.M2n = ∆2z(n) . (6)We also have [28]:f2n = Fd eψψdz(n)dn∆2(M2n)d eψψ−2 , (7)with Fd eψψ a function depending on the scaling dimensionof the operator as well as the details of the underlyingdynamics [15]. Because of the presence of the fictitiousmass terms the potential reads:V = mΛγmU Nf∞∑n=1fnϕn + m̄ΛγmU Nf∞∑n=1fnϕ̄n+∞∑n=1M2nϕnϕ̄n . (8)The bar over the fields and the fermion mass indicatescomplex conjugation. The extremum condition yields:〈ϕ̄n〉 = −mΛγmU NffnM2n, (9)yielding:〈O〉 = −m̄ΛγmU Nf∞∑n=1f2nM2n. (10)We now take the limit ∆2 → 0 and the sum becomes anintegral. For any specific function z(n) it is easy to showthat:〈O〉 = −m̄ΛγmU NfFd eψψΩ [ΛUV ,ΛIR] , (11)withΩ [ΛUV ,ΛIR] =11− γm[Λ2(1−γm)UV − Λ2(1−γm)IR]. (12)The ultraviolet and infrared cutoffs are introduced totame the integral in the respective regions. A simplephysical interpretation of these cutoffs is the following.At very high energies, at scales above ΛU , the under-lying theory flows to the ultraviolet fixed point and wehave to abandon the description in terms of the com-posite operator. This immediately suggests that ΛUV isnaturally identified with ΛU . The presence of the massterm induces a mass gap, which is the quantity we aretrying to determine. The induced physical mass gap is anatural infrared cutoff. We, hence, identify ΛIR with thephysical value of the condensate. We find:〈ψ̃fc ψcf 〉 ∝ −mΛ2U , 0 < γm < 1 , (13)〈ψ̃fc ψcf 〉 ∝ −mΛ2U logΛ2U|〈O〉|, γm → 1 , (14)〈ψ̃fc ψcf 〉 ∝ −m3−γm1+γm Λ4γm1+γmU , 1 < γm ≤ 2 . (15)We used 〈ψ̃ψ〉 ∼ ΛγmU 〈O〉 to relate the expectation valueof O to the one of the fermion condensate. Via an allowedaxial rotation m is now real and positive. It is instructiveto compare these results with the ones obtained via naivedimensional analysis (NDA) [29] also discussed in [30]and in [15]. We find:〈ψ̃fc ψcf 〉NDA ∝ −m3−γm1+γm Λ4γm1+γmU . (16)Note that one recovers the previous scaling as function ofm (up to logarithmic corrections) only for 1 ≤ γm ≤ 2.The failure of NDA for a smaller anomalous dimension isdue to the fact that the ultraviolet physics is not capturedby NDA [15].Instantons: The underlying gauge theory suffers of anaxial anomaly resulting in an explicit breaking of theUA(1) symmetry. To take into account this phenomenonwe add the following instanton-induced term to the po-tential of the theory [31, 32, 33, 34, 35, 36, 37, 38, 39, 40]:cΛ4Udet[O eψψ]Λd eψψ NfU+ h.c. = cONfΛ(d eψψ Nf−4)U+ h.c. . (17)In terms of the fields ϕm the VEV equation reads:〈ϕ`〉 = −f`M2`[m̄NfΛγmU + c̄ Nf(∑n fn〈ϕ̄n〉)Nf−1Λd eψψ Nf−4U].(18)We search for a solution of the previous equation of theform 〈ϕ`〉 = af`M2`. Substituting in the previous expres-sion we deduce:āNf−1C̄ + a+ M̄ = 0 , (19)3withC̄ =c̄NfΛd eψψ Nf−4U(Fd eψψΩ [ΛU ,ΛIR])Nf−1, (20)M̄ = m̄NfΛγmU . (21)We have already taken the ∆ → 0 limit in (20). Weanalytically solve for a in the two extreme cases, i.e., noinstanton contribution (c = 0):|a| = |M | , δa = π(1 + 2k)− δM with c = 0 .(22)δa and δM are the phases respectively of a and M . Thissolution was found above. In the limit in which the in-stanton term dominates over the linear term in a thesolution is:|a||M |=(|M |2−Nf|C|) 1Nf−1, δa =δM − δc − π(1− 2k)Nf − 1.(23)k is an integer. In the instanton dominated (ID) limit:|〈O〉ID| =(|M |Λd eψψNf−4u|c|Nf) 1Nf−1. (24)The Fd eψψΩ [ΛU ,ΛIR] term cancels. Explicitating themass term dependence:〈ψ̃fc ψcf 〉ID ∝(mΛU) 1Nf−1Λ3U . (25)This ID contribution dominates for large values of thefermion mass .In the small mass regime we can solve (19) perturba-tively in the mass. This expansion is well defined for0 < γm ≤ 1 since here the C coefficient is not affected bythe IR divergence. To the next leading term in m:〈O〉 ' −(M̄ + (−M)Nf−1 C̄) Fd eψψΩ [ΛUV ,ΛIR] . (26)The introduction of θ-term is a source of strong CPviolation appearing at low energies via the identificationδM = δm + ω and δc = θ +Nfω with δm the phase ofthe fermion mass and ω an overall axial rotation. Us-ing ω one can rotate away one of the two phases butnot both simultaneously. The net result is the pres-ence in the action of θeff = θ − Nfδm and we find〈O〉ID = Exp[−iθeff − π(1− 2k)Nf − 1]|〈O〉ID|. According tothe all-order beta function [9] an anomalous dimensionsmaller than one requires, for fermions in the fundamen-tal representation, a number of flavors larger than 11N/3.Even for N = 2 the number of flavors needed is suffi-ciently large to predict that the instanton corrections arenegligible. In the case of fermionic matter transform-ing according to higher dimensional representations theeffects of the instantons are more relevant since fewerflavors are needed to reach the conformal window [4] re-ducing the exponent of the instanton induced term. Be-sides, for any representation, in the region 1 < γm ≤ 2,instantons are relevant as we shall demonstrate below.2 Dirac Fermions, 2-index Symmetric Represen-tation of SU(3): There are lattice indications that thistheory may develop an IRFP [21]. The instanton inducedpotential term here has the lowest possible exponent, i.e.Nf = 2. The all-order beta function predicts γm = 1.3.This is the regime where the instanton term cannot beneglected. We obtain:〈O〉 =2meiδaΛγmU Fd eψψΩ [ΛUV ,ΛIR]cos δa +2|c| cos(δa+δc)Λ2d eψψ−4UFd eψψΩ [ΛUV ,ΛIR], (27)where m > 0, θeff = 0, δM = π and sin δa = |C| sin(δa +δc) determines δa. For γm > 1 Ω is IR dominated leadingto:〈ψ̃f{c1,c2}ψ{c1,c2}f 〉 = −mcΛ2U +O(m2) , (28)rather than〈ψ̃f{c1,c2}ψ{c1,c2}f 〉 ∝ −(mΛU) 3−γm1+γmΛ3U , (29)valid for 1 < γm ≤ 2 but without the instanton term.Without using the information of the beta function onemight still imagine the possibility that the anomalousdimension is smaller than one. In this case Ω is UVdivergent, we replace ΛUV with ΛU and find that thedependence of the condensate on the mass is still linear.2 Dirac Fermions in the Adjoint Representationof SU(2): This theory is also being investigated on thelattice [19, 20, 22, 23, 24]. Here the instanton inducedterm has 2Nf = 4 as exponent and one can still solveanalytically for the condensate. If the underlying confor-mal theory possesses an IRFP according to the all-orderbeta function γm = 3/4 then the condensate has a lineardependence on the fermion mass. It has a dependenceproportional to m13 for larger values of the mass.Conformal Pions: At any nonzero value of the fermionmass the chiral and conformal symmetries are explicitlybroken and single particle states emerge at low energies.The relevant ones here are the conformal pions, i.e. thewould be Goldstones which in the limit of zero fermionmass cannot be described via single particle states. Weidentify them via〈O eψψf ′f 〉 = 〈O〉U with U = ei πFπ . (30)π = πaT a and T a are the set of broken generators nor-malized according to Tr[T aT b]= δab1/2. Substituting4(30) in (3) and expanding up to the second order in thepion fields we have:m2πF2π = −mΛγmU 〈O[m]〉 . (31)Having determined the dependence on m of 〈O[m]〉 theabove generalizes the similar one in QCD [41, 42, 43]known as the Gell-Mann Oakes Renner (GMOR) rela-tion. This relation can be used to discover and classify,in a physical way, different conformal fixed points. Forexample for the theories investigated above we expect,for a very small fermion mass, m2πF2π = m2Λ2U . At largermasses the scaling is different for the three cases and itcan be easily deduced from our results. A similar ef-fective Lagrangian was introduced in [15]. Assume nowthat the underlying gauge theory has not developed anIRFP. In this case there are only two possibilities: i) chi-ral symmetry breaks spontaneously yielding a condensatewhose leading term in m is a constant; ii) chiral sym-metry is intact but a scale is still generated. Chirallypaired partners emerge together with massless compos-ite fermions appearing to saturate the ’t Hooft anomalymatching conditions. One can investigate the finite vol-ume effects using the conformal pion lagrangian in theε-regime [44].We presented a novel analysis of the nonperturbativephysics related to the chiral dynamics of theories pos-sessing an IRFP. The results are testable via first prin-ciples lattice simulations. Deviations from the QCD-likeGMOR relation can be used to disentangle conformal dy-namics from non-conformal dynamics. The low energyeffective theories presented here, for the conformal pi-ons, can be extended to describe dynamical breaking ofthe electroweak symmetry featuring nearly conformal dy-namics such as MWT and UMT. The signals from MWTat the LHC are being investigated while UMT also leadsto interesting dark matter candidates [8].∗ Electronic address: sannino@ifk.sdu.dk[1] F. Sannino, arXiv:0804.0182 [hep-ph].[2] S. Weinberg, Phys. Rev. D 19, 1277 (1979).[3] L. Susskind, Phys. Rev. D 20, 2619 (1979).[4] F. Sannino and K. Tuominen, Phys. Rev. D 71, 051901(2005)[5] D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev.D 72, 055001 (2005)[6] D. D. Dietrich and F. Sannino, Phys. Rev. D 75, 085018(2007)[7] R. Foadi, M. T. Frandsen, T. A. Ryttov and F. Sannino,Phys. Rev. D 76, 055005 (2007)[8] T. A. Ryttov and F. Sannino, arXiv:0809.0713 [hep-ph].To appear in Physical Review D.[9] T. A. Ryttov and F. Sannino, Phys. Rev. D 78, 065001(2008)[10] E. Eichten and K. D. Lane, Phys. Lett. B 90, 125 (1980).[11] B. Holdom, Phys. Rev. D 24, 1441 (1981).[12] K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev.Lett. 56, 1335 (1986).[13] T. W. Appelquist, D. Karabali and L. C. R. Wijeward-hana, Phys. Rev. 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Tuominen, PoS LATTICE2008, 065 (2008)[arXiv:0810.3722 [hep-lat]].[25] T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev.Lett. 100, 171607 (2008).[26] A. Deuzeman, M. P. Lombardo and E. Pallante,arXiv:0804.2905 [hep-lat].[27] Z. Fodor, K. Holland, J. Kuti, D. Nogradi andC. Schroeder, arXiv:0809.4890 [hep-lat].[28] M. A. Stephanov, Phys. Rev. D 76, 035008 (2007).[29] P. J. Fox, A. Rajaraman and Y. Shirman, Phys. Rev. D76, 075004 (2007).[30] M. A. Luty, arXiv:0806.1235 [hep-ph].[31] E. Witten, Nucl. Phys. B 156, 269 (1979).[32] G. Veneziano, Nucl. Phys. B 159, 213 (1979).[33] E. Witten, Annals Phys. 128, 363 (1980).[34] C. Rosenzweig, J. Schechter and C. G. Trahern, Phys.Rev. D 21, 3388 (1980).[35] P. Di Vecchia and G. Veneziano, Nucl. Phys. B 171, 253(1980).[36] P. Nath and R. L. Arnowitt, Phys. Rev. D 23, 473 (1981).[37] K. Kawarabayashi and N. Ohta, Nucl. Phys. B 175, 477(1980).[38] M. A. Shifman and A. I. Vainshtein, Nucl. Phys. B 277,456 (1986)[39] G. ’t Hooft, Phys. 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Conformal Chiral Dynamics

       University of Southern DenmarkConformal Chiral DynamicsSannino, FrancescoPublished in:Physical Review DDOI:10.1103/PhysRevD.80.017901Publication date:2009Citation for pulished version (APA):Sannino, F. (2009). Conformal Chiral Dynamics. Physical Review D, D80, 017901.https://doi.org/10.1103/PhysRevD.80.017901Go 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. 2025Conformal Chiral DynamicsFrancesco Sannino∗High Energy Center, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark.We investigate the chiral dynamics of gauge theories developing an infrared stable fixed point. Wedetermine the dependence of the bilinear fermion condensate on the underlying fermion mass andits anomalous dimension. We introduce the instanton contributions and investigate how they affectthe dynamics near the fixed point. We generalize the Gell-Mann Oakes Renner relation and suggestto use it to uncover the presence of an infrared fixed point of the underlying gauge theory. Ourresults have an immediate impact on the construction of sensible extensions of the Standard Modelof particle interactions and the general understanding of the phase diagram of strongly coupledtheories.Non-abelian gauge theories exist in a number of dis-tinct phases which can be classified according to the char-acteristic dependence of the potential energy on the dis-tance between two well separated static sources. Thecollection of all of these different behaviors, when repre-sented, for example, in the flavor-color space, constitutesthe Phase Diagram of the given gauge theory.Knowing the phase diagram of strongly coupled theo-ries has an immediate impact on the construction of sen-sible extensions of the Standard Model (SM) of particleinteractions [1]. Dynamical breaking of the electroweaksymmetry is a time-honored example [2, 3]. The use offermions transforming according to higher dimensionalrepresentations of the new gauge group is leading to sev-eral interesting phenomenological possibilities [4, 5, 6]such as Minimal Walking Technicolor (MWT)[7] and Ul-tra Minimal Walking Technicolor (UMT) [8]. These mod-els lie close, in theory space, to theories with non-trivialinfrared fixed points (IRFP)s [4, 9]. In the vicinity ofsuch a zero of the beta-function the coupling constantflows slowly, i.e. walks [10, 11, 12, 13]. Knowledge of thephase diagram is relevant also to provide natural ultra-violet completions of unparticle [14] models [1, 15]. Togain analytic insight one can use the conjectured all-orderbeta function for nonsupersymmetric gauge theories [9]together with the constraints from the unitarity of theconformal operators [16]. Other approaches are based onthe older truncated Schwinger-Dyson approach (SD) [17]or more recent ideas [18]. The analytical phase diagramobtained by this approach, and a comparison of it to re-cent lattice results [19, 20, 21, 22, 23, 24, 25, 26, 27], issummarized in [1].The goal here is to study the chiral properties at theIRFP, i.e. the conformal chiral dynamics.Conformal Chiral Dynamics: Our starting point is anonsupersymmetric non-abelian gauge theory with suf-ficient massless fermionic matter to develop a nontriv-ial IRFP. The cartoon of the running of the couplingconstant is represented in Fig. 1. In the plot ΛU isthe dynamical scale below which the IRFP is essentiallyreached. It can be defined as the scale for which α is2/3 of the fixed point value in a given renormalizationscheme. If the theory possesses an IRFP the chiral con-�Energy�U�*FIG. 1: Running of the coupling constant in an asymptoti-cally free gauge theory developing an infrared fixed point fora value α = α∗.densate must vanish at large distances. Here we want tostudy the behavior of the condensate when a flavor sin-glet mass term is added to the underlying Lagrangian:∆L = −mψ̃ψ + h.c. , (1)with m the fermion mass and ψfc as well as ψ̃cf left trans-forming two component spinors, c and f represent colorand flavor indices. The omitted color and flavor indices,in the Lagrangian term, are contracted. We consider firstthe case of fermionic matter in the fundamental represen-tation of the SU(N) gauge group. We then generalize ourresults to the case of higher dimensional representations.The effect of such a term is to break the conformalsymmetry together with some of the global symmetriesof the underlying gauge theory. The composite operator:O eψψf ′f = ψ̃f ′ψf , (2)has mass dimension d eψψ = 3− γm with γm the anoma-lous dimension of the mass term. At the fixed point γmis a positive number smaller than two [16]. We assumem ΛU . Dimensional analysis demands:∆L→ −mΛγmU Tr[O eψψ] + h.c. . (3)The mass term is a relevant perturbation around theIRFP driving the theory away from the fixed point.arXiv:0811.0616v3  [hep-ph]  2 Feb 20092It will induce a nonzero vacuum expectation value forO eψψ itself proportional to δf ′f . It is convenient to defineTr[O eψψ] = NfO with O a flavor singlet operator. Therelevant low energy Lagrangian term is then:−mΛγmU NfO + h.c. . (4)To determine the vacuum expectation value of O we re-place it, formally, with a sum over an infinite number ofcanonically normalized single particle states [28]:O(x) =∞∑n=1fnϕn(x) . (5)Each state possesses a mass Mn whose value is controlledby an artificial mass gap ∆ and a function of n, call itz(n), with the properties z(n+ 1) > z(n) and z(1) = 1.M2n = ∆2z(n) . (6)We also have [28]:f2n = Fd eψψdz(n)dn∆2(M2n)d eψψ−2 , (7)with Fd eψψ a function depending on the scaling dimensionof the operator as well as the details of the underlyingdynamics [15]. Because of the presence of the fictitiousmass terms the potential reads:V = mΛγmU Nf∞∑n=1fnϕn + m̄ΛγmU Nf∞∑n=1fnϕ̄n+∞∑n=1M2nϕnϕ̄n . (8)The bar over the fields and the fermion mass indicatescomplex conjugation. The extremum condition yields:〈ϕ̄n〉 = −mΛγmU NffnM2n, (9)yielding:〈O〉 = −m̄ΛγmU Nf∞∑n=1f2nM2n. (10)We now take the limit ∆2 → 0 and the sum becomes anintegral. For any specific function z(n) it is easy to showthat:〈O〉 = −m̄ΛγmU NfFd eψψΩ [ΛUV ,ΛIR] , (11)withΩ [ΛUV ,ΛIR] =11− γm[Λ2(1−γm)UV − Λ2(1−γm)IR]. (12)The ultraviolet and infrared cutoffs are introduced totame the integral in the respective regions. A simplephysical interpretation of these cutoffs is the following.At very high energies, at scales above ΛU , the under-lying theory flows to the ultraviolet fixed point and wehave to abandon the description in terms of the com-posite operator. This immediately suggests that ΛUV isnaturally identified with ΛU . The presence of the massterm induces a mass gap, which is the quantity we aretrying to determine. The induced physical mass gap is anatural infrared cutoff. We, hence, identify ΛIR with thephysical value of the condensate. We find:〈ψ̃fc ψcf 〉 ∝ −mΛ2U , 0 < γm < 1 , (13)〈ψ̃fc ψcf 〉 ∝ −mΛ2U logΛ2U|〈O〉|, γm → 1 , (14)〈ψ̃fc ψcf 〉 ∝ −m3−γm1+γm Λ4γm1+γmU , 1 < γm ≤ 2 . (15)We used 〈ψ̃ψ〉 ∼ ΛγmU 〈O〉 to relate the expectation valueof O to the one of the fermion condensate. Via an allowedaxial rotation m is now real and positive. It is instructiveto compare these results with the ones obtained via naivedimensional analysis (NDA) [29] also discussed in [30]and in [15]. We find:〈ψ̃fc ψcf 〉NDA ∝ −m3−γm1+γm Λ4γm1+γmU . (16)Note that one recovers the previous scaling as function ofm (up to logarithmic corrections) only for 1 ≤ γm ≤ 2.The failure of NDA for a smaller anomalous dimension isdue to the fact that the ultraviolet physics is not capturedby NDA [15].Instantons: The underlying gauge theory suffers of anaxial anomaly resulting in an explicit breaking of theUA(1) symmetry. To take into account this phenomenonwe add the following instanton-induced term to the po-tential of the theory [31, 32, 33, 34, 35, 36, 37, 38, 39, 40]:cΛ4Udet[O eψψ]Λd eψψ NfU+ h.c. = cONfΛ(d eψψ Nf−4)U+ h.c. . (17)In terms of the fields ϕm the VEV equation reads:〈ϕ`〉 = −f`M2`[m̄NfΛγmU + c̄ Nf(∑n fn〈ϕ̄n〉)Nf−1Λd eψψ Nf−4U].(18)We search for a solution of the previous equation of theform 〈ϕ`〉 = af`M2`. Substituting in the previous expres-sion we deduce:āNf−1C̄ + a+ M̄ = 0 , (19)3withC̄ =c̄NfΛd eψψ Nf−4U(Fd eψψΩ [ΛU ,ΛIR])Nf−1, (20)M̄ = m̄NfΛγmU . (21)We have already taken the ∆ → 0 limit in (20). Weanalytically solve for a in the two extreme cases, i.e., noinstanton contribution (c = 0):|a| = |M | , δa = π(1 + 2k)− δM with c = 0 .(22)δa and δM are the phases respectively of a and M . Thissolution was found above. In the limit in which the in-stanton term dominates over the linear term in a thesolution is:|a||M |=(|M |2−Nf|C|) 1Nf−1, δa =δM − δc − π(1− 2k)Nf − 1.(23)k is an integer. In the instanton dominated (ID) limit:|〈O〉ID| =(|M |Λd eψψNf−4u|c|Nf) 1Nf−1. (24)The Fd eψψΩ [ΛU ,ΛIR] term cancels. Explicitating themass term dependence:〈ψ̃fc ψcf 〉ID ∝(mΛU) 1Nf−1Λ3U . (25)This ID contribution dominates for large values of thefermion mass .In the small mass regime we can solve (19) perturba-tively in the mass. This expansion is well defined for0 < γm ≤ 1 since here the C coefficient is not affected bythe IR divergence. To the next leading term in m:〈O〉 ' −(M̄ + (−M)Nf−1 C̄) Fd eψψΩ [ΛUV ,ΛIR] . (26)The introduction of θ-term is a source of strong CPviolation appearing at low energies via the identificationδM = δm + ω and δc = θ +Nfω with δm the phase ofthe fermion mass and ω an overall axial rotation. Us-ing ω one can rotate away one of the two phases butnot both simultaneously. The net result is the pres-ence in the action of θeff = θ − Nfδm and we find〈O〉ID = Exp[−iθeff − π(1− 2k)Nf − 1]|〈O〉ID|. According tothe all-order beta function [9] an anomalous dimensionsmaller than one requires, for fermions in the fundamen-tal representation, a number of flavors larger than 11N/3.Even for N = 2 the number of flavors needed is suffi-ciently large to predict that the instanton corrections arenegligible. In the case of fermionic matter transform-ing according to higher dimensional representations theeffects of the instantons are more relevant since fewerflavors are needed to reach the conformal window [4] re-ducing the exponent of the instanton induced term. Be-sides, for any representation, in the region 1 < γm ≤ 2,instantons are relevant as we shall demonstrate below.2 Dirac Fermions, 2-index Symmetric Represen-tation of SU(3): There are lattice indications that thistheory may develop an IRFP [21]. The instanton inducedpotential term here has the lowest possible exponent, i.e.Nf = 2. The all-order beta function predicts γm = 1.3.This is the regime where the instanton term cannot beneglected. We obtain:〈O〉 =2meiδaΛγmU Fd eψψΩ [ΛUV ,ΛIR]cos δa +2|c| cos(δa+δc)Λ2d eψψ−4UFd eψψΩ [ΛUV ,ΛIR], (27)where m > 0, θeff = 0, δM = π and sin δa = |C| sin(δa +δc) determines δa. For γm > 1 Ω is IR dominated leadingto:〈ψ̃f{c1,c2}ψ{c1,c2}f 〉 = −mcΛ2U +O(m2) , (28)rather than〈ψ̃f{c1,c2}ψ{c1,c2}f 〉 ∝ −(mΛU) 3−γm1+γmΛ3U , (29)valid for 1 < γm ≤ 2 but without the instanton term.Without using the information of the beta function onemight still imagine the possibility that the anomalousdimension is smaller than one. In this case Ω is UVdivergent, we replace ΛUV with ΛU and find that thedependence of the condensate on the mass is still linear.2 Dirac Fermions in the Adjoint Representationof SU(2): This theory is also being investigated on thelattice [19, 20, 22, 23, 24]. Here the instanton inducedterm has 2Nf = 4 as exponent and one can still solveanalytically for the condensate. If the underlying confor-mal theory possesses an IRFP according to the all-orderbeta function γm = 3/4 then the condensate has a lineardependence on the fermion mass. It has a dependenceproportional to m13 for larger values of the mass.Conformal Pions: At any nonzero value of the fermionmass the chiral and conformal symmetries are explicitlybroken and single particle states emerge at low energies.The relevant ones here are the conformal pions, i.e. thewould be Goldstones which in the limit of zero fermionmass cannot be described via single particle states. Weidentify them via〈O eψψf ′f 〉 = 〈O〉U with U = ei πFπ . (30)π = πaT a and T a are the set of broken generators nor-malized according to Tr[T aT b]= δab1/2. Substituting4(30) in (3) and expanding up to the second order in thepion fields we have:m2πF2π = −mΛγmU 〈O[m]〉 . (31)Having determined the dependence on m of 〈O[m]〉 theabove generalizes the similar one in QCD [41, 42, 43]known as the Gell-Mann Oakes Renner (GMOR) rela-tion. This relation can be used to discover and classify,in a physical way, different conformal fixed points. Forexample for the theories investigated above we expect,for a very small fermion mass, m2πF2π = m2Λ2U . At largermasses the scaling is different for the three cases and itcan be easily deduced from our results. A similar ef-fective Lagrangian was introduced in [15]. Assume nowthat the underlying gauge theory has not developed anIRFP. 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Conformal Chiral Dynamics

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Syddansk Universitets Forskerportal

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