Amato, A.

Rantalaiho, T.

Rummukainen, K.

Tuominen, K.

Tähtinen, S.

arXiv

Helsingin yliopiston digitaalinen arkisto

Approaching the conformal window: systematicstudy of the particle spectrum in SU(2) field theorywith N f = 2, 4 and 6.Alessandro Amato,Helsinki Institute of Physics and University of HelsinkiE-mail: alessandro.amato@helsinki.fiTeemu Rantalaiho,Helsinki Institute of Physics and University of HelsinkiE-mail: teemu.rantalaiho@helsinki.fiKari Rummukainen,Helsinki Institute of Physics and University of HelsinkiE-mail: kari.rummukainen@helsinki.fiKimmo Tuominen,Helsinki Institute of Physics and University of HelsinkiE-mail: kimmo.i.tuominen@helsinki.fiSara Tähtinen∗Helsinki Institute of Physics and University of HelsinkiE-mail: sara.tahtinen@helsinki.fiIt is expected that SU(2) gauge theory with N f fundamental fermions has an infrared fixed pointwhen N f is between ∼ 6 and 10. We study the hadron spectrum and scale setting in SU(2) gaugefield theory with N f = 2,4,6 using hypercubic stout smeared Wilson-clover (HEX) action. Thecase N f = 2 is QCD-like, whereas N f = 6 is close to the lower edge of the conformal window. Inour study the length scales are determined by using the gradient flow approach.The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/arXiv:1511.04947v1  [hep-lat]  16 Nov 2015Approaching the conformal window Sara Tähtinen1. IntroductionIn Technicolor (TC) theories electroweak symmetry is dynamically broken by a chiral conden-sate of new strong dynamics consisting of SU(N) gauge field theory with N f massless fermions.The Higgs sector of the SM then arises as the effective low energy composite description of thisnew strong dynamics solving the naturalness and fine-tuning problems of the SM. However, theTC paradigm alone does not explain SM fermion masses, which are typically assumed to arisefrom extended TC (ETC) interactions. Absence of flavour changing neutral current interactionsand compatibility with the precision electroweak measurements suggest that the coupling constantshould walk, i.e. remain almost constant g∼ g∗ over a large scale separation.At large enough value of N f the theory looses asymptotic freedom while at low values of N fthe chiral symmetry is broken as in QCD. These are two examples of phases of gauge theories,characterized by the evolution of the gauge coupling, and passage from one another can be studiedby changing the value of N f . Between these two phases, as a function of N f , lies the so calledconformal window: the range of values of N f for which the β -function β = µ dgdµ of the theoryreaches zero at g2 = g2∗. Then the theory has an infrared fixed point (IRFP), leading to scaleinvariant long distance behaviour. Walking behaviour is expected to occur at the values of N fdirection below the boundary between the conformal window and the phase where the theory breakschiral symmetry. As the lower boundary of the conformal window lies at strong coupling, latticesimulations are needed to determine the properties of these theories.In this work we study the particle spectrum of SU(2) gauge theory with N f = 2, 4 and 6fundamental representation fermions. In up to 4-loop perturbation theory, all of these theories havean IRFP [1]; however, for N f = 6 the fixed point appears at very strong coupling. On the otherhand, if one estimates the onset of the chiral symmetry breaking using a ladder approximation, theN f = 6 theory is predicted to fall within the conformal window [2, 3]. On the lattice, there is clearevidence for chiral symmetry breaking for N f ≤ 4, and for the existence of an IRFP at N f = 8 and10 [4,5]. However, for N f = 6 a clear picture has not yet emerged, with inconclusive and conflictingresults in the literature [4, 6–9].2. Mass spectrumIn figs. 1 to 3 the pion and rho masses are presented as functions of the quark mass. At N f = 2the pions behave exactly as expected in QCD-like chiral symmetry breaking scenario, i.e. pionsbecome massless ∝ √mQ. Because of the familiarity of this scenario we do not attempt to reachvery small quark masses.However, already in the N f = 4 case the small quark mass region becomes problematic: thehadron masses in the small mQ region deviate from the expected chiral behaviour by apparentlybecoming ∼ constant and having a finite intercept as mQ → 0. This behaviour is due to finitevolume effects, as can be observed from the right panel in fig. 2: the “levelling off” happens atheavier quark masses in 243×48 lattices than in 323×60 ones. The sensitivity to the finite volumecan be explained by the rapid growth in the hadron sizes as mQ approaches zero, as will be discussedbelow in connection with the scale setting.2Approaching the conformal window Sara Tähtinen0 0.1 0.2 0.3 0.4am PCAC00.20.40.60.811.2am pi  mass ρ  masssquare root fitFigure 1: The hadron masses at N f = 2 with β = 1.0. The square root behaviour of the pseudoscalaris evident.0 0.05 0.1 0.15 0.2 0.25amPCAC00.20.40.60.8am pi mass,  β=1.0 ρ mass,  β=1.0 pi mass,  β=0.8 ρ mass,  β=0.80 0.05 0.1 0.15 0.2amPCAC00.20.40.60.8amsquare root fit ρ mass, 24x48 pi mass, 24x48 ρ mass, 32x60 pi mass, 32x60Figure 2: Left: Pion and rho masses at N f = 4 with β = 1 and 0.8. The points with yellow centerrefer to the bigger lattice size (323x60). Right: the lattice size effect to the hadron masses atβ = 0.8.3Approaching the conformal window Sara Tähtinen0 0.1 0.2amPCAC00.20.40.60.81am ρ mass,  β=0.6 pi mass,  β=0.6 ρ mass,  β=0.7 ρ mass,  β =0.7 ρ mass,  β=0.8 pi mass,  β=0.80 0.1 0.2amPCAC00.20.40.60.81am ρ mass, 24x48 pi mass, 24x48 ρ mass, 32x60 pi mass, 32x60Figure 3: Left: Pion and rho masses in N f = 6 with β = 0.6, 0.7 and 0.8. The points with yellowcenter refer to the bigger lattice size. Right: The effect of the lattice size on the hadron masses withβ = 0.8.As expected, the finite volume effects can be avoided to some degree by using larger volume atfixed lattice spacing (which soon becomes very expensive) or by using coarser lattices. The latticespacing dependence is clearly visible on the left panel of fig. 2, where the levelling off happens atheavier quarks at β = 1 than at β = 0.8. However, even at the largest volumes and coarsest latticespacing we are not yet able to verify the expected mpi ∝ m1/2Q behaviour.The results at N f = 6 are still preliminary. However, we can see that for N f = 6 the finitevolume effects are even stronger than for N f = 4, and it is difficult to reach pion massess which aresmaller than mpi ' 0.4 even at coarsest lattices and largest lattice volumes we use. This indicatesthat the growth in the size of the hadrons as mQ→ 0 is very rapid.3. Scale setting: gradient flowThe scale setting procedure in lattice QCD is needed in order to relate the lattice scale to somephysically known quantity. Popular choices have been rho mass, string tension and the Sommerscale r0 [13], or the Sommer-type scale r1 [14], but presently the gradient flow approach [15] hasbecome the preferred one. Even though it is an artificial way of setting the scale, it is known tobe very precise, cheap and straightforward to implement. In order to find the gradient flow lengthscale√t0/a, one has to solve t0 fromt20 〈E(t0)〉= 0.3, (3.1)where E(t) = 14Gaµν(t)Gaµν(t) is the action density, with Gaµν(t) the lattice field strength at flowtime t [15]. We use here the Symanzik-improved lattice field strength. A related observable w0 isdefined via [16]tddt(t2 〈E(t)〉)|t=w20 = 0.3, (3.2)4Approaching the conformal window Sara Tähtinen0.0 0.1 0.2 0.3 0.4 0.5amPCAC02468101214√ t0/aNf = 2, β = 1.0Nf = 4, β = 0.8Nf = 4, β = 1.0Nf = 6, β = 0.6Nf = 6, β = 0.7Nf = 6, β = 0.8Figure 4: Results of the√t0-scale. Yellow circles and diamonds refer to the results obtained withthe bigger lattice.0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7mPCAC√t01.01.11.21.31.41.51.6mρ/mpiNf = 2, β = 1.0Nf = 4, β = 0.8Nf = 4, β = 1.0Nf = 6, β = 0.6Nf = 6, β = 0.7Nf = 6, β = 0.8Figure 5: Results in physical units. The results are observed to settle on a universal curve; theunderlying reason for this scaling is currently unclear.5Approaching the conformal window Sara Tähtinenbut in practice these two scales do not differ much. Here we present results using the t0-scale.In theories in or near the conformal window the scale setting takes another role: if the theoryhas an IRFP it is scale invariant in the infrared. Now the scale setting procedure can be used tomeasure the approach to conformality as the conformal symmetry breaking quark mass mQ→ 0. Ifthe theory has an IRFP we expect that√t0/a diverges in this limit.In fig. 4 the scale√t0/a is presented as a function of amQ. It can be seen that at N f = 2 thescale is more or less independent of mQ, but at N f = 4 and 6 it grows rapidly at small amQ. Whilethe N f = 4 -curves seem to have a finite intecept at mq = 0, the behaviour seen at N f = 6 is notobvious. Because√t0/a is a measure of the hadron size, strong finite-volume effects at small mQcan be expected. If we use√t0 to fix the scale, then bigger√t0/a value means smaller latticespacing a, and for fixed amPCAC the quarks are heavier. In other words, the bigger N f and thevalue of β are, the harder it is to reach to the small physical quark mass region, because the latticespacing a gets smaller and smaller. Thus, near the conformal window very strong lattice coupling(i.e small β ) is needed in order to make mQ small, and to avoid finite volume effects.In fig. 5 we plot the mρ/mpi mass ratio against the “physical” quark mass mQ√t0. Remarkably,all of our measurements fall on an “universal” curve, independent of N f or the lattice spacing. ForN f = 2 the ratio diverges, as expected, as mQ → 0. The N f = 4 case follows N f = 2, but muchmore slowly. Clearly, small β (large lattice spacing and large physical volume) is required to reachlarge mρ/mpi . The behaviour of the N f = 6 data is interesting: at β = 0.8 and 0.7 the points remainesentially fixed at mQ√t0 = 0.6 and 0.4, respectively, independent of mQa, and only at β = 0.6 thepoints slowly move towards smaller mQ√t0 as mQa is lowered. This behaviour may indicate almostconformal behaviour: in a system which becomes conformal as mQ→ 0 the quark masses are theonly physical scale. Hence mQ√t0 can be a constant. However, if the system is not quite conformalat mQ = 0, at light enough quark masses the chiral symmetry breaking behaviour is revealed. Thisrequires large physical volumes, i.e. small β . This may indicate that N f = 6 is indeed outside ofthe conformal window. In the future we will clarify this behaviour further by studying the N f = 8case, which we know has an IRFP [5, 17].4. ConclusionsWe have initiated a systematic study of the spectrum in SU(2) gauge theory with N f = 2, 4and 6 fundamental representation fermions, i.e. when approaching the conformal window. In ourstudy, we have focused on the hadron spectrum and the gradient flow scale-setting.For the mass spectrum, our central finding is that the finite volume effects are stronger thanexpected, making it difficult to reach the chiral small quark mass regime even at N f = 4. In orderto reach small quark masses we are forced to use small β (strong bare coupling) in order for thephysical volume to be large enough.Considering the scale setting, our main observation is that amQ dependence of the√t0/a scalebecomes stronger as N f or β is increased. By plotting the mass ratio mρ/mpi against mQ√t0, where√t0 is the scale obtained from gradient flow, we observed that all our measurements fall on anuniversal curve. This may indicate that the chiral symmetry is broken when N f ≤ 6, i.e. N f = 6 isstill outside of the conformal window.6Approaching the conformal window Sara Tähtinen5. AcknowledgmentsThis work is supported by the Academy of Finland grants 267842 and 267286. S.T. and T.R.acknowledge support from the Magnus Ehrnrooth Foundation. The simulations have been done atthe Finnish IT Center for Science (CSC).References[1] T. van Ritbergen, J. A. M. Vermaseren, and S. A. Larin. The Four loop beta function in quantumchromodynamics. Phys. Lett., B400:379–384, 1997.[2] Thomas Appelquist, John Terning, and L. C. R. Wijewardhana. The Zero temperature chiral phasetransition in SU(N) gauge theories. Phys. Rev. Lett., 77:1214–1217, 1996.[3] Francesco Sannino and Kimmo Tuominen. Orientifold theory dynamics and symmetry breaking.Phys. Rev., D71:051901, 2005.[4] Tuomas Karavirta, Jarno Rantaharju, Kari Rummukainen, and Kimmo Tuominen. Determining theconformal window: SU(2) gauge theory with N f = 4, 6 and 10 fermion flavours. JHEP, 05:003, 2012.[5] Viljami Leino, Tuomas Karavirta, Jarno Rantaharju, Teemu Rantalaiho, Kari Rummukainen, Joni M.Suorsa, and Kimmo Tuominen. 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Approaching the conformal window: Systematic study of the particle spectrum in SU(2) field theory with Nf = 2, 4 and 6

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