We propose an alternative paradigm to the conjectured Miransky scaling
potentially underlying the physics describing the transition from the
conformally broken to the conformally restored phase when tuning certain
parameters such as the number of flavors in gauge theories. According to the
new paradigm the physical scale and henceforth also the massive spectrum of the
theory jump at the lower boundary of the conformal window. In particular we
propose that a theory can suddenly jump from a Quantum Chromodynamics type
spectrum, at the lower boundary of the conformal window, to a conformal one
without particle interpretation. The jumping scenario, therefore, does not
support a near-conformal dynamics of walking type. We will also discuss the
impact of jumping dynamics on the construction of models of dynamical
electroweak symmetry breaking.Comment: 4 pages. 6 Figure

Sannino, Francesco

English

arXiv

 We propose an alternative paradigm to the conjectured Miransky scaling potentially underlying the physics describing the transition from the conformally broken to the conformally restored phase when tuning certain parameters such as the number of flavors in gauge theories. According to the new paradigm the physical scale and henceforth also the massive spectrum of the theory jump at the lower boundary of the conformal window. In particular, we propose that a theory can suddenly jump from a Quantum Chromodynamics (QCD) type spectrum, at the lower boundary of the conformal window, to a conformal one without particle interpretation. The jumping scenario, therefore, does not support a near-conformal dynamics of walking type. We will argue that jumping dynamics can occur for relevant gauge theories such as super QCD and QCD. We will also discuss the impact of jumping dynamics on the construction of models of dynamical electroweak symmetry breaking. </jats:p

FRANCESCO SANNINO

Crossref

Modern Physics Letters A

JUMPING DYNAMICS

We propose an alternative paradigm to the conjectured Miransky scaling potentially underlying the physics describing the transition from the conformally broken to the conformally restored phase when tuning certain parameters such as the number of flavors in gauge theories. According to the new paradigm the physical scale and henceforth also the massive spectrum of the theory jump at the lower boundary of the conformal window. In particular, we propose that a theory can suddenly jump from a Quantum Chromodynamics (QCD) type spectrum, at the lower boundary of the conformal window, to a conformal one without particle interpretation. The jumping scenario, therefore, does not support a near-conformal dynamics of walking type. We will argue that jumping dynamics can occur for relevant gauge theories such as super QCD and QCD. We will also discuss the impact of jumping dynamics on the construction of models of dynamical electroweak symmetry breaking.</p

Syddansk Universitets Forskerportal

       University of Southern DenmarkJumping DynamicsSannino, FrancescoPublished in:Modern Physics Letters ADOI:10.1142/S0217732313501277Publication date:2013Citation for pulished version (APA):Sannino, F. (2013). Jumping Dynamics. Modern Physics Letters A, 28(30), Article 1350127.https://doi.org/10.1142/S0217732313501277Go 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. 2025Jumping DynamicsFrancesco Sanninor∗r CP3-Origins and the Danish Institute for Advanced Study,University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark.We propose an alternative paradigm to the conjectured Miransky scaling potentially underlyingthe physics describing the transition from the conformally broken to the conformally restored phasewhen tuning certain parameters such as the number of flavors in gauge theories. According to thenew paradigm the physical scale and henceforth also the massive spectrum of the theory jump at thelower boundary of the conformal window. In particular we propose that a theory can suddenly jumpfrom a Quantum Chromodynamics type spectrum, at the lower boundary of the conformal window,to a conformal one without particle interpretation. The jumping scenario, therefore, does not supporta near-conformal dynamics of walking type. We will also discuss the impact of jumping dynamics onthe construction of models of dynamical electroweak symmetry breaking.Preprint: CP3-Origins-2012-12 & DIAS-2012-13Understanding the non-perturbative dynamics ofgauge theories of fundamental interactions constitutesa formidable challenge. Recently a considerable efforthas been made to unveil the long distance conformaldynamics of these theories, see [1] for a recent review.However, despite much efforts, we do not yet knowthe physical properties at the boundary between a con-formally broken and a conformally restored phase forgeneric gauge theories. This problem remains an impor-tant mystery to solve. A famous conjecture has been putforward some time ago [2, 3] and it is known as Miranskyscaling. Here we will argue for the potential existenceof another intriguing possibility leading to a radicallydifferent near-conformal behavior. For reader’s conve-nience and for setting up the stage we start with a brief,but self-contained, review of the the Miransky scalingand modeling. We will then introduce the alternativescenario and deduce the consequences for models of dy-namical electroweak symmetry breaking.MIRANSKY SCALING AND WALKING DYNAMICSThis scaling arises under the following assumptions: i)A given gauge theory possesses simultaneously, at least,a non-trivial infrared (IR) fixed point and an ultraviolet(UV) one; ii) Upon changing an external parameter ofthe theory, e.g. the number of flavors, at a critical valueof this parameter the IR fixed point merges with the UVfixed point; iii) This merging is sufficiently smooth thatthe nearby conformal phase is felt, in the conformallybroken phase, for values of the external parameter nearthe phase transition.Without loss of generality it is possible to model thebeta function near the critical number of flavors as fol-lows:βMY = −α2(α − 1 −√δ) (α − 1 +√δ)= −α2((α− 1)2 − δ) .(1)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-0.3-0.2-0.10.00.1ΑΒMY∆=0.1∆=0∆=-0.1∆=-0.2FIG. 1. βMY for different values of δ = n f − ncf .The double zero at the origin embodies asymptotic free-dom and δ = n f − ncf . For positive values of δ the betafunction possesses a non-trivial IR and UV fixed point atthe following values of the coupling:αIR = 1 −√δ , and αUV = 1 +√δ . (2)At n f = ncf the fixed points merge and for n f < ncf the betafunction looses the non-trivial fixed points. For negativeδ within the following range:−18< δ ≤ 0 (3)there is a global maximum of the beta function at theorigin, a local minimum at α = 14 (3−√1 + 8δ) and a localmaximum at α = 14 (3 +√1 + 8δ). For illustration we plotthe beta function for different values of δ in Fig. 1. It ispossible to find an analytical solution to the RG equation:d lnµ =dαβMY, (4)arXiv:1205.4246v1  [hep-ph]  18 May 20122reading:lnµµ0=α(1 + δ)ArcTanh[−1+α√δ]+√δ(1 − δ + α ln[(α−1)2−δα2])α(−1 + δ)2√δ∣∣∣∣α(µ)α(µ0).(5)If we were to consider the case of δ positive, but smallerthan unity so that asymptotic freedom is kept, we woulddiscover that there are three distinct branches. The oneto the left of the IR fixed point, the one where α is inbetween the nontrivial IR and UV fixed point, and theone to the right of the nontrivial UV fixed point. To theleft of the IR fixed point one starts the flow from anyµ0 sufficiently close to the trivial UV fixed point andone ends up at the attractive IR fixed point. Anotherasymptotically safe theory is defined in between the twonon-trivial fixed points. In this region the coupling runsat low energies to the IR fixed point and raises at highenergies till the non-trivial UV fixed point is reached.Finally, to the right of the non-trivial UV fixed point thetheory, and hence beta function, runs in the deep infraredto increasingly large values of the coupling.We turn now our attention to negative values of δcorresponding to the phase in which the beta func-tion features no non-trivial fixed points. To elucidatethe near-conformal dynamics we investigate the region−1/8 < δ < 0. In particular we will consider the limit−δ = ncf − n f → 0. In the deep infrared the couplingconstant runs to infinity and we start the running in theUV near α = 1 at µ0. With these boundary conditions wefind:ΛMY =µ0ncf − n fexp− π2 √ncf − n f , n f → ncf , n f ≤ ncf .(6)Here ΛMY is the infrared scale to be identified, for exam-ple, with a physical scale of the theory such as the massof a hadron. This scale vanishes exponentially fast whenapproaching the critical number of flavors above whichthe infrared fixed point is generated. This exponentialbehavior is the essence of the Miransky scaling.In Fig. 2 we plot the running of the coupling constantfor different negative values of δ with the normalizationcondition α(µ0) = 0.1. The figure visualizes the ideaof walking dynamics introduced by Holdom [4, 5] andfurther crystallized in [6, 7]. In lay terms the couplingconstant runs slowly, i.e. walks, towards the infraredvalue remaining near constant over a range of energiesbecoming wider and wider as one approaches, as func-tion of δ, the double fixed point. Further assuming thatthe beta function corresponds to an underlying gaugetheory featuring fermions we now determine the scal-ing behavior of the chiral condensate of the theory inthe walking regime. Defining with γ the anomalous di-mension of the mass of the Dirac fermion Q in a given-120 -100 -80 -60 -40 -20 00.00.20.40.60.81.01.21.4ln ΜΑHΜL∆=-18∆=-0.01∆=-0.002∆=-0.001∆=0.0005FIG. 2. Running of the coupling constant coming from βMY fordifferent values of δ = n f − ncf within the walking regime. Allthe solutions are normalized at µ0 such that α(µ0) = 0.1 and µis normalized to µ0.representation of an underlying gauge group we havethe following well-known RG equation:〈Q̄Q〉µ = exp(∫ µΛd(lnµ)γ(α(µ)))〈Q̄Q〉Λ= exp(∫ α(µ)α(Λ)dαγ(α)β(α))〈Q̄Q〉Λ (7)relating the condensate at two different energies. UsingβMY case in the walking region we have〈Q̄Q〉µ = exp(∫ α(µ)α(Λ)dαγ(α)−α2((α − 1)2 + |δ|))〈Q̄Q〉Λ ' exp(γ(1)∫ α(µ)α(Λ)dα1βMY)〈Q̄Q〉Λ =( µΛ)γ(1)〈Q̄Q〉Λ . (8)In the last passage we have used the definition of thebeta function and, in the first step, assumed that theanomalous dimension of the mass operator is smoothacross the phase transition. γ(1) = γ(αIR = αUV) is thevalue of the anomalous dimension at the merger. Wehave re-derived the power-law enhancement of the chi-ral condensate with the energy distinctive of walking dy-namics. Since γ is evaluated at the fixed point its value3is scheme-independent [8]. If we further model γ = αwe have γ(αIR/UV) = 1 ∓√δ , with γ(αIR) + γ(αUV) = 2.JUMPING DYNAMICSThe previous section embodies the standard paradigmof walking dynamics. However this picture is far fromestablished analytically or via first principle lattice sim-ulations. It is therefore relevant to consider other the-oretical scenarios and their impact on particle physicsphenomenology. We start by observing that there is thelogical possibility that the full beta function of the theorydevelops, at least, a zero in the denominator. This oc-curs exactly for supersymmetric gauge theories [9] andthe all-orders beta function conjectured to be valid alsofor non-supesymmetric gauge theories with fermionicmatter [8, 10]. Moreover it is reasonable to expect thatthe full perturbative and non-perturbative contributionsto the beta function conspire to generate a non-trivialpole structure [11]. Whatever the pole structure is, if theunderlying theory displays conformality, there will bealso zeros in the numerator of the beta function associ-ated to the non-trivial fixed point structure of the theory.Here we consider the simplest example in which the betafunction has a simple nontrivial zero in the numeratorand a simple pole. We will always assume the existenceof the trivial double zero at the origin so that the betafunction contains information about the asymptoticallyfree nature of the theory. Without loss of generality wewrite:βJump = −α2 1 − δ − α1 − α. (9)By construction this beta function has a zero in the nu-merator for any δ which is to the left of the pole valueαpole = 1 for δ positive and to the right for δ negative.Here we take again δ = n f − ncf . It is straightforwardto show that this zero corresponds to an IR (UV) fixedpoint for δ > 0 (δ < 0) :αIR(UV) = 1 − δ , δ > 0 (δ < 0) and |δ| ≤ 1 . (10)Because of the presence of the pole the beta function de-scribes two disconnected theories. One which is contin-uously connected to the asymptotically free underlyinggauge theory and the other which is not. We plot thebeta function for positive and negative values of deltain Fig. 3. At exactly δ = 0 the numerator and denom-inator of the beta function cancel and we are left withβδ=0Jump = −α2 which is the red curve in Fig. 3. What hap-pens at the phase boundary? We will demonstrate thatthere is a sudden jump as we drop the number of flavorsbelow the critical number of flavors (i.e. at δ = 0) of theintrinsic physical scale of the theory.We start by constructing the analytical solution for theRG equation of the coupling which reads:lnµµ0=1α(1 − δ)∣∣∣∣α(µ)α(µ0)+δ(1 − δ)2ln[1 − δ − αα] ∣∣∣∣α(µ)α(µ0). (11)Holding fixed, as done for the Miransky scaling case, thecoupling constant at a given renormalization scale oneobserves that the newly generated scale increases withdecreasing the number of flavors below the conformalwindow in the following way:ΛJump = Λc[1 − (ncf − n f ) ln(ncf − n f)], n f → ncf , n f ≤ ncf .(12)Λc = µ0 exp[lnα0α0]is the renormalization group invari-ant scale of the theory at the critical number of flavors.However for n f > ncf no infrared scale is generated andnecessarily there must be a jump in the spectrum fromΛc to zero. This result shows that βMY and βJump de-scribe two distinct physical systems. For illustration wesummarize in Fig. 5 the behavior of the physical scaleof the theory, as function of number of flavors, for Mi-ransky scaling and jumping dynamics. To compare thetwo scaling laws we normalized the two scales at a givenvalue of n f .Can jumping dynamics be used for models of dynamicalelectroweak symmetry breaking? At the conformal bound-ary the dynamics is QCD like and therefore one observesonly a logarithmic enhancement of the condensate of thetype 〈Q̄Q〉µ ' γ(1) ln(µΛ)〈 Q̄Q〉Λ. The jumping dynam-ics does not lead to power-law enhancement of the chiralcondensate required for walking Technicolor. Further-more the S-parameter in the jumping scenario automati-cally respects the lower bound put forward in [12] giventhat, opportunely normalized, at the phase boundary isas small as the one for QCD. Henceforth the answer to theoriginal question is that one can break the electroweaktheory via jumping dynamics but cannot accommodatethe generation of the standard model fermion massesfollowing the walking paradigm nor drastically reducethe QCD-like S-parameter.We have shown that it is possible to devise a sim-ple framework according to which the approach to thelong distance conformality does not display any sign ofwalking dynamics. All the current lattice investigationsof the conformal window are not able to differentiatewalking from jumping. The reasons being that: i) Thelattice results, for the moment, are performed for a fixednumber of flavors and therefore either there is a nonzeroinfrared scale or the theory is conformal; ii) The precisedetermination of the chiral condensate is not a simpletask making harder to disentangle a power-law from alogarithmic enhancement of the condensate as functionof the renormalization scale as well as the number of fla-vors; iii) Even if the underlying dynamics is of walkingtype (with or without the introduction of four-fermion40.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3-2-101ΑΒJump∆=0∆=-0.05∆=-0.20.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-4-3-2-1012ΑΒJump∆=0∆=0.05∆=0.2FIG. 3. βJump for different values of δ = n f − ncf .interactions) the extension of the region in the numberof flavors and four-fermion coupling is not known andmight be tiny; iv) Measuring a large anomalous dimen-sion of the mass is encouraging but alone insufficient todemonstrate the existence of walking dynamics.-10 -8 -6 -4 -2 00.00.20.40.60.81.01.2ln ΜΑHΜL∆=0.1∆=0.05∆=0∆=-0.2∆=-0.1∆=-0.05FIG. 4. Running of the coupling constant due to βJump fordifferent values of δ = n f −ncf . All the solutions are normalizedat µ0 such that α(µ0) = 0.1, and of course µ is normalized to µ0.0.80 0.85 0.90 0.95 1.00 1.050.00.20.40.60.81.01.21.41-n fn fcL@nfDLJumpLMYFIG. 5. Mass gap dependence on the number of flavors forthe Miransky scaling (lower black curve) and for the jumpingdynamics (upper blue curve).Because of the discontinuity of the order parameterat the conformal phase transition, i.e. of the vacuumexpectation value of the trace of the improved energymomentum tensor which is proportional to the intrinsicscale of the theory, jumping dynamics corresponds to afirst order conformal phase transition. First order phasetransitions are common in nature and therefore we ex-pect jumping dynamics to constitute a likely scenariowith inevitable important consequences on a large num-ber of research fields ranging from a better understand-ing of strong dynamics and its holographic engineeringto the construction of sensible extensions of the standardmodel.I thank O. Antipin, A. Hietanen, M. Mojaza, C. Pica,T. Ryttov, J. Schechter and U. Sondegaard for useful dis-cussions and careful reading of the manuscript.∗ sannino@cp3.dias.sdu.dk[1] F. Sannino, Acta Phys. Polon. B 40, 3533 (2009)[arXiv:0911.0931 [hep-ph]].[2] V. A. Miransky and K. Yamawaki, Mod. Phys. Lett. A 4,129 (1989).[3] V. A. Miransky and K. Yamawaki, Phys. Rev. D 55,5051 (1997) [Erratum-ibid. D 56, 3768 (1997)] [arXiv:hep-th/9611142].[4] B. Holdom, Phys. Rev. D 24, 1441 (1981).[5] B. Holdom, Phys. Lett. B 150, 301 (1985).[6] K. Yamawaki, M. Bando and K. -i. Matumoto, Phys. Rev.Lett. 56, 1335 (1986).[7] T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana,Phys. Rev. Lett. 57, 957 (1986).[8] C. Pica and F. Sannino, Phys. Rev. D 83, 116001 (2011)[arXiv:1011.3832 [hep-ph]].[9] V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Za-kharov, Nucl. Phys. B 229, 381 (1983).[10] T. A. Ryttov and F. Sannino, Phys. Rev. D 78, 065001 (2008)[arXiv:0711.3745 [hep-th]].[11] F. A. Chishtie, V. Elias, V. A. Miransky and T. G. Steele,Prog. Theor. Phys. 104, 603 (2000) [hep-ph/9905291].[12] F. Sannino, Phys. Rev. D 82, 081701 (2010) [arXiv:1006.0207[hep-lat]].

Jumping Dynamics

       University of Southern DenmarkJumping DynamicsSannino, FrancescoPublished in:Modern Physics Letters ADOI:10.1142/S0217732313501277Publication date:2013Citation for pulished version (APA):Sannino, F. (2013). Jumping Dynamics. Modern Physics Letters A, 28(30), Article 1350127.https://doi.org/10.1142/S0217732313501277Go 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. 2025Jumping DynamicsFrancesco Sanninor∗r CP3-Origins and the Danish Institute for Advanced Study,University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark.We propose an alternative paradigm to the conjectured Miransky scaling potentially underlyingthe physics describing the transition from the conformally broken to the conformally restored phasewhen tuning certain parameters such as the number of flavors in gauge theories. According to thenew paradigm the physical scale and henceforth also the massive spectrum of the theory jump at thelower boundary of the conformal window. In particular we propose that a theory can suddenly jumpfrom a Quantum Chromodynamics type spectrum, at the lower boundary of the conformal window,to a conformal one without particle interpretation. The jumping scenario, therefore, does not supporta near-conformal dynamics of walking type. We will also discuss the impact of jumping dynamics onthe construction of models of dynamical electroweak symmetry breaking.Preprint: CP3-Origins-2012-12 & DIAS-2012-13Understanding the non-perturbative dynamics ofgauge theories of fundamental interactions constitutesa formidable challenge. Recently a considerable efforthas been made to unveil the long distance conformaldynamics of these theories, see [1] for a recent review.However, despite much efforts, we do not yet knowthe physical properties at the boundary between a con-formally broken and a conformally restored phase forgeneric gauge theories. This problem remains an impor-tant mystery to solve. A famous conjecture has been putforward some time ago [2, 3] and it is known as Miranskyscaling. Here we will argue for the potential existenceof another intriguing possibility leading to a radicallydifferent near-conformal behavior. For reader’s conve-nience and for setting up the stage we start with a brief,but self-contained, review of the the Miransky scalingand modeling. We will then introduce the alternativescenario and deduce the consequences for models of dy-namical electroweak symmetry breaking.MIRANSKY SCALING AND WALKING DYNAMICSThis scaling arises under the following assumptions: i)A given gauge theory possesses simultaneously, at least,a non-trivial infrared (IR) fixed point and an ultraviolet(UV) one; ii) Upon changing an external parameter ofthe theory, e.g. the number of flavors, at a critical valueof this parameter the IR fixed point merges with the UVfixed point; iii) This merging is sufficiently smooth thatthe nearby conformal phase is felt, in the conformallybroken phase, for values of the external parameter nearthe phase transition.Without loss of generality it is possible to model thebeta function near the critical number of flavors as fol-lows:βMY = −α2(α − 1 −√δ) (α − 1 +√δ)= −α2((α− 1)2 − δ) .(1)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-0.3-0.2-0.10.00.1ΑΒMY∆=0.1∆=0∆=-0.1∆=-0.2FIG. 1. βMY for different values of δ = n f − ncf .The double zero at the origin embodies asymptotic free-dom and δ = n f − ncf . For positive values of δ the betafunction possesses a non-trivial IR and UV fixed point atthe following values of the coupling:αIR = 1 −√δ , and αUV = 1 +√δ . (2)At n f = ncf the fixed points merge and for n f < ncf the betafunction looses the non-trivial fixed points. For negativeδ within the following range:−18< δ ≤ 0 (3)there is a global maximum of the beta function at theorigin, a local minimum at α = 14 (3−√1 + 8δ) and a localmaximum at α = 14 (3 +√1 + 8δ). For illustration we plotthe beta function for different values of δ in Fig. 1. It ispossible to find an analytical solution to the RG equation:d lnµ =dαβMY, (4)arXiv:1205.4246v1  [hep-ph]  18 May 20122reading:lnµµ0=α(1 + δ)ArcTanh[−1+α√δ]+√δ(1 − δ + α ln[(α−1)2−δα2])α(−1 + δ)2√δ∣∣∣∣α(µ)α(µ0).(5)If we were to consider the case of δ positive, but smallerthan unity so that asymptotic freedom is kept, we woulddiscover that there are three distinct branches. The oneto the left of the IR fixed point, the one where α is inbetween the nontrivial IR and UV fixed point, and theone to the right of the nontrivial UV fixed point. To theleft of the IR fixed point one starts the flow from anyµ0 sufficiently close to the trivial UV fixed point andone ends up at the attractive IR fixed point. Anotherasymptotically safe theory is defined in between the twonon-trivial fixed points. In this region the coupling runsat low energies to the IR fixed point and raises at highenergies till the non-trivial UV fixed point is reached.Finally, to the right of the non-trivial UV fixed point thetheory, and hence beta function, runs in the deep infraredto increasingly large values of the coupling.We turn now our attention to negative values of δcorresponding to the phase in which the beta func-tion features no non-trivial fixed points. To elucidatethe near-conformal dynamics we investigate the region−1/8 < δ < 0. In particular we will consider the limit−δ = ncf − n f → 0. In the deep infrared the couplingconstant runs to infinity and we start the running in theUV near α = 1 at µ0. With these boundary conditions wefind:ΛMY =µ0ncf − n fexp− π2 √ncf − n f , n f → ncf , n f ≤ ncf .(6)Here ΛMY is the infrared scale to be identified, for exam-ple, with a physical scale of the theory such as the massof a hadron. This scale vanishes exponentially fast whenapproaching the critical number of flavors above whichthe infrared fixed point is generated. This exponentialbehavior is the essence of the Miransky scaling.In Fig. 2 we plot the running of the coupling constantfor different negative values of δ with the normalizationcondition α(µ0) = 0.1. The figure visualizes the ideaof walking dynamics introduced by Holdom [4, 5] andfurther crystallized in [6, 7]. In lay terms the couplingconstant runs slowly, i.e. walks, towards the infraredvalue remaining near constant over a range of energiesbecoming wider and wider as one approaches, as func-tion of δ, the double fixed point. Further assuming thatthe beta function corresponds to an underlying gaugetheory featuring fermions we now determine the scal-ing behavior of the chiral condensate of the theory inthe walking regime. Defining with γ the anomalous di-mension of the mass of the Dirac fermion Q in a given-120 -100 -80 -60 -40 -20 00.00.20.40.60.81.01.21.4ln ΜΑHΜL∆=-18∆=-0.01∆=-0.002∆=-0.001∆=0.0005FIG. 2. Running of the coupling constant coming from βMY fordifferent values of δ = n f − ncf within the walking regime. Allthe solutions are normalized at µ0 such that α(µ0) = 0.1 and µis normalized to µ0.representation of an underlying gauge group we havethe following well-known RG equation:〈Q̄Q〉µ = exp(∫ µΛd(lnµ)γ(α(µ)))〈Q̄Q〉Λ= exp(∫ α(µ)α(Λ)dαγ(α)β(α))〈Q̄Q〉Λ (7)relating the condensate at two different energies. UsingβMY case in the walking region we have〈Q̄Q〉µ = exp(∫ α(µ)α(Λ)dαγ(α)−α2((α − 1)2 + |δ|))〈Q̄Q〉Λ ' exp(γ(1)∫ α(µ)α(Λ)dα1βMY)〈Q̄Q〉Λ =( µΛ)γ(1)〈Q̄Q〉Λ . (8)In the last passage we have used the definition of thebeta function and, in the first step, assumed that theanomalous dimension of the mass operator is smoothacross the phase transition. γ(1) = γ(αIR = αUV) is thevalue of the anomalous dimension at the merger. Wehave re-derived the power-law enhancement of the chi-ral condensate with the energy distinctive of walking dy-namics. Since γ is evaluated at the fixed point its value3is scheme-independent [8]. If we further model γ = αwe have γ(αIR/UV) = 1 ∓√δ , with γ(αIR) + γ(αUV) = 2.JUMPING DYNAMICSThe previous section embodies the standard paradigmof walking dynamics. However this picture is far fromestablished analytically or via first principle lattice sim-ulations. It is therefore relevant to consider other the-oretical scenarios and their impact on particle physicsphenomenology. We start by observing that there is thelogical possibility that the full beta function of the theorydevelops, at least, a zero in the denominator. This oc-curs exactly for supersymmetric gauge theories [9] andthe all-orders beta function conjectured to be valid alsofor non-supesymmetric gauge theories with fermionicmatter [8, 10]. Moreover it is reasonable to expect thatthe full perturbative and non-perturbative contributionsto the beta function conspire to generate a non-trivialpole structure [11]. Whatever the pole structure is, if theunderlying theory displays conformality, there will bealso zeros in the numerator of the beta function associ-ated to the non-trivial fixed point structure of the theory.Here we consider the simplest example in which the betafunction has a simple nontrivial zero in the numeratorand a simple pole. We will always assume the existenceof the trivial double zero at the origin so that the betafunction contains information about the asymptoticallyfree nature of the theory. Without loss of generality wewrite:βJump = −α2 1 − δ − α1 − α. (9)By construction this beta function has a zero in the nu-merator for any δ which is to the left of the pole valueαpole = 1 for δ positive and to the right for δ negative.Here we take again δ = n f − ncf . It is straightforwardto show that this zero corresponds to an IR (UV) fixedpoint for δ > 0 (δ < 0) :αIR(UV) = 1 − δ , δ > 0 (δ < 0) and |δ| ≤ 1 . (10)Because of the presence of the pole the beta function de-scribes two disconnected theories. One which is contin-uously connected to the asymptotically free underlyinggauge theory and the other which is not. We plot thebeta function for positive and negative values of deltain Fig. 3. At exactly δ = 0 the numerator and denom-inator of the beta function cancel and we are left withβδ=0Jump = −α2 which is the red curve in Fig. 3. What hap-pens at the phase boundary? We will demonstrate thatthere is a sudden jump as we drop the number of flavorsbelow the critical number of flavors (i.e. at δ = 0) of theintrinsic physical scale of the theory.We start by constructing the analytical solution for theRG equation of the coupling which reads:lnµµ0=1α(1 − δ)∣∣∣∣α(µ)α(µ0)+δ(1 − δ)2ln[1 − δ − αα] ∣∣∣∣α(µ)α(µ0). (11)Holding fixed, as done for the Miransky scaling case, thecoupling constant at a given renormalization scale oneobserves that the newly generated scale increases withdecreasing the number of flavors below the conformalwindow in the following way:ΛJump = Λc[1 − (ncf − n f ) ln(ncf − n f)], n f → ncf , n f ≤ ncf .(12)Λc = µ0 exp[lnα0α0]is the renormalization group invari-ant scale of the theory at the critical number of flavors.However for n f > ncf no infrared scale is generated andnecessarily there must be a jump in the spectrum fromΛc to zero. This result shows that βMY and βJump de-scribe two distinct physical systems. For illustration wesummarize in Fig. 5 the behavior of the physical scaleof the theory, as function of number of flavors, for Mi-ransky scaling and jumping dynamics. To compare thetwo scaling laws we normalized the two scales at a givenvalue of n f .Can jumping dynamics be used for models of dynamicalelectroweak symmetry breaking? At the conformal bound-ary the dynamics is QCD like and therefore one observesonly a logarithmic enhancement of the condensate of thetype 〈Q̄Q〉µ ' γ(1) ln(µΛ)〈 Q̄Q〉Λ. The jumping dynam-ics does not lead to power-law enhancement of the chiralcondensate required for walking Technicolor. Further-more the S-parameter in the jumping scenario automati-cally respects the lower bound put forward in [12] giventhat, opportunely normalized, at the phase boundary isas small as the one for QCD. Henceforth the answer to theoriginal question is that one can break the electroweaktheory via jumping dynamics but cannot accommodatethe generation of the standard model fermion massesfollowing the walking paradigm nor drastically reducethe QCD-like S-parameter.We have shown that it is possible to devise a sim-ple framework according to which the approach to thelong distance conformality does not display any sign ofwalking dynamics. All the current lattice investigationsof the conformal window are not able to differentiatewalking from jumping. The reasons being that: i) Thelattice results, for the moment, are performed for a fixednumber of flavors and therefore either there is a nonzeroinfrared scale or the theory is conformal; ii) The precisedetermination of the chiral condensate is not a simpletask making harder to disentangle a power-law from alogarithmic enhancement of the condensate as functionof the renormalization scale as well as the number of fla-vors; iii) Even if the underlying dynamics is of walkingtype (with or without the introduction of four-fermion40.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3-2-101ΑΒJump∆=0∆=-0.05∆=-0.20.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-4-3-2-1012ΑΒJump∆=0∆=0.05∆=0.2FIG. 3. βJump for different values of δ = n f − ncf .interactions) the extension of the region in the numberof flavors and four-fermion coupling is not known andmight be tiny; iv) Measuring a large anomalous dimen-sion of the mass is encouraging but alone insufficient todemonstrate the existence of walking dynamics.-10 -8 -6 -4 -2 00.00.20.40.60.81.01.2ln ΜΑHΜL∆=0.1∆=0.05∆=0∆=-0.2∆=-0.1∆=-0.05FIG. 4. Running of the coupling constant due to βJump fordifferent values of δ = n f −ncf . All the solutions are normalizedat µ0 such that α(µ0) = 0.1, and of course µ is normalized to µ0.0.80 0.85 0.90 0.95 1.00 1.050.00.20.40.60.81.01.21.41-n fn fcL@nfDLJumpLMYFIG. 5. Mass gap dependence on the number of flavors forthe Miransky scaling (lower black curve) and for the jumpingdynamics (upper blue curve).Because of the discontinuity of the order parameterat the conformal phase transition, i.e. of the vacuumexpectation value of the trace of the improved energymomentum tensor which is proportional to the intrinsicscale of the theory, jumping dynamics corresponds to afirst order conformal phase transition. First order phasetransitions are common in nature and therefore we ex-pect jumping dynamics to constitute a likely scenariowith inevitable important consequences on a large num-ber of research fields ranging from a better understand-ing of strong dynamics and its holographic engineeringto the construction of sensible extensions of the standardmodel.I thank O. Antipin, A. Hietanen, M. Mojaza, C. Pica,T. Ryttov, J. Schechter and U. Sondegaard for useful dis-cussions and careful reading of the manuscript.∗ sannino@cp3.dias.sdu.dk[1] F. Sannino, Acta Phys. Polon. B 40, 3533 (2009)[arXiv:0911.0931 [hep-ph]].[2] V. A. Miransky and K. Yamawaki, Mod. Phys. Lett. A 4,129 (1989).[3] V. A. Miransky and K. Yamawaki, Phys. Rev. D 55,5051 (1997) [Erratum-ibid. D 56, 3768 (1997)] [arXiv:hep-th/9611142].[4] B. Holdom, Phys. Rev. D 24, 1441 (1981).[5] B. Holdom, Phys. Lett. B 150, 301 (1985).[6] K. Yamawaki, M. Bando and K. -i. Matumoto, Phys. Rev.Lett. 56, 1335 (1986).[7] T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana,Phys. Rev. Lett. 57, 957 (1986).[8] C. Pica and F. Sannino, Phys. Rev. D 83, 116001 (2011)[arXiv:1011.3832 [hep-ph]].[9] V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Za-kharov, Nucl. Phys. B 229, 381 (1983).[10] T. A. Ryttov and F. Sannino, Phys. Rev. D 78, 065001 (2008)[arXiv:0711.3745 [hep-th]].[11] F. A. Chishtie, V. Elias, V. A. Miransky and T. G. Steele,Prog. Theor. Phys. 104, 603 (2000) [hep-ph/9905291].[12] F. Sannino, Phys. Rev. D 82, 081701 (2010) [arXiv:1006.0207[hep-lat]].

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Jumping Dynamics

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