JHEP05(2009)025 D,E He Adjoint May 6, 2009 April 2, 2009 Ries, Renor- 2

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JHEP05(2009)025 d,e he adjoint May 6, 2009 April 2, 2009 ries, Renor- 2. However, : : March 11, 2009 ∼ : December 17, 2008 L : β , Published Accepted Revised to similar measurements the physical quark mass most conformal in the in- Received viour at sively confirm the existence ons of a change from chiral and Kimmo Tuominen ossible phase diagram of this his theory and present results 10.1088/1126-6708/2009/05/025 b,c e of these results for the future doi: ty of Helsinki, Finland Published by IOP Publishing for SISSA Kari Rummukainen b [email protected] [email protected] , , Jarno Rantaharju, 0812.1467 a Technicolor and Composite Models, Lattice Gauge Field Theo An SU(2) gauge theory with two fermions transforming under t [email protected] SISSA 2009 Department of Physics, FloridaCP International 204, University, 11200 SW 8thepartment of St, Physics, Miami, University FLP.O. of 33199, Box Oulu, U.S.A. 3000, FI-90014 UniversityDepartment of of Physics, Oulu, University Finland P.O. of Box Helsinki, 64, FI-00014 UniversityDepartment of of Physics, Helsinki, University Finland P.O. of Box Jyväskylä, 35, FI-40014 UniversityHelsinki Finland of Institute Jyväskylä, of Physics, P.O. Box 64, katu Gustaf 2, Hällströmin FI-00014 Universi E-mail: [email protected] b c e d a c

Abstract: Ari J. Hietanen, malization Group, Nonperturbative Effects ArXiv ePrint: representation of the gaugefrared. group We may use lattice appear simulationson conformal to the study or the al masses spectrumdetermined of of t through several the gauge axialsymmetry singlet Ward breaking identity states to and as a find a phase indicati consistent function with of conformal beha Spectrum of SU(2) latticeadjoint gauge Dirac theory flavours with two the measurement of the spectrumof is conformal not fixed alone point sufficient inwith to fundamental this deci fermions. theory Based as on welattice the show results theory we by and sketch comparing a discuss p measurement the of applicability the and evolution importanc of the coupling constant. Keywords: JHEP05(2009)025 ] - – 1 5 7 2 β 5 14 ]. These theories 10 – 8 nd the theory appears ch either feature an in- n species (techniquarks), oes, originates from tech- d at very short distances, ied is important for model ndard Model gauge bosons ly weakly to the Standard damental scalar particle [ ered. Conformal symmetry models. These models were e some striking implications stant is, in a wide range of TC models, based on a tech- e theories was provided in [ een known for some time that ady manifest in the Standard l currents due to the extended heories [ infrared fixed points of the like strongly interacting theory, ed point at strong coupling. e phase structure of these theories eudo-Goldstone bosons due to the ]. In this regard, an especially interesting 1 – 1 – ]. Recently theoretical aspects of these unparticles 3 ], where they have been considered as arising in a 4 Another phenomenological motivation to study theories whi ]. The Higgs sector in these theories consists of a new fermio function. This means thatit while becomes the a couplingconformal. constant runs One when over of probe some the energy phenomenological connections range to in thes the infrared a 7 1 Introduction and summary Different phases of non-Abelian gauge field theories are alre 3 Results 4 Conclusions and outlook model of elementary particle interactions, andas, charting th for example, numbersbuilding of beyond colours the and Standard quark model, flavours see are e.g. var [ Contents 1 Introduction and summary 2 The model and lattice formulation are nearly conformal,energy, i.e. governed the by an evolution attractive of quasi-stable infrared the fix coupling con where the possibilityModel of through a effective fully operatorsdetermines conformal at the sector low mass dimensions energies coupledfor of was on the these consid related operators low which energyhave hav observables [ been further investigated in [ charged under a newnicolor gauge sector interaction straightforwardly (technicolor). extrapolated from Early had a several QCD- problems. These includetechnicolor flavour interactions changing and neutra unwanted additional lightbreaking ps of the chiral symmetrythese of the problems techniquarks. are It solved has in b so called walking technicolor t natural way within a strongly coupled gauge theory. frared fixed point ornicolor are, (TC) in and theory thedevised space, associated a close extended long to technicolor time (ETC) one agoand to which fundamental explain fermions d the without mass the patterns need of to the introduce a Sta fun class of theories are quantum field theories with nontrivial JHEP05(2009)025 1 ) = (1.1) small R ( T becomes -function , and the = 4 SYM ab 1 1 β δ β -function is /β β N 0 . The form of ]. This theory, β 1 and 2 undamental rep- 14 π . This zero corre- R , -function never ac- 1 16 f β ) gauge field theory ntiquark condensate − N ]. However, it is more ) N = R ith infrared stable fixed gure g-wavelength probes are 12 or example ntually the ( 2 on-abelian gauge theories. n a generic asymptotically ∗ T ) g rom the asymptotically free bative values of the coupling ed. Supersymmetry provides y close proximity to it allows ch decouples the quarks from R ermions. the antiscreening contribution construction of theories which ( , 2 2 ] ) -function vanishes identically as C 2 4 β 11 5 π ) representation g ) gauge field theory with fermions, − e [ small and positive, while ) = 0 at N ∗ N = 0. However, imagine adding enough f (16 0 g 1 β g N ( ) β β R − ( 2 T , ], and later to all orders in [ ) 3 π f g – 2 – G 13 N -function are sketched in figure 16 ( ) 2 β 0 R β C ( − 3 T 20 3 4 -function negative. Therefore the -function for SU( − β )= β − 2 g ) ) ( β G G ( ( 2 2 -function of a non-supersymmetric SU( . Nevertheless, we can discuss the features shown in figure 1 C C β 0, which guarantees asymptotic freedom. Perturbative 3 3 11 34 . Another way to destabilize the infrared fixed point is to add < 1 ) = = g 0 1 ( β β is small, perturbative analysis can be applied [ β ∗ g ) is the second Casimir invariant for SU( ). As is well known for QCD, for modest amount of matter in the f R ( b R 2 T C a Namely, one has to take into account the formation of quark-a Therefore, there is need for concrete examples of theories w Several possible generic forms of a R However, for a recent conjecture about its possible form, se T 1 -function is of the form shown by the dashed-dotted curve in fi matter in the fundamental representation to make by considering the perturbative tr( remains unknown to date sponds to an attractiveside infrared the fixed theory point. becomesconsidered. If conformal, If approached very f unlikelikely QCD, that when such infrared lon fixedand points one do has not to appearThis account at can for pertur lead full to nonperturbative the walking dynamics dynamics. of n points but not many concrete examples are currently known. F however, is conformal atenough all structure scales to and allow not for just nonperturbative in analysis the and infrar the full nonperturbative leading to the spontaneous breakinginfrared of dynamics chiral and symmetry, keeps whi the explicit chiral symmetry breaking, e.g. a mass term for the f where the coefficients are defined up to two loop order as and tually reaches the would-bethe infrared fixed coupling point, to but “walk” the slowly ver over a wide range of scales. Eve resentation there is not enoughof screening the to gluons compensate and for negative. Then there appears a nontrivial zero, β becomes zero only in the ultraviolet fixed point at provides a concrete realization of a theory where the was first shown on the two loop level in [ repelled from the proximityfree of theory, the see fixed figure point and behaves as i JHEP05(2009)025 ed line, range of lattice o fermions in the ], little is known in 2 15 g ning the behaviour of the lattice previously in he lattice using e.g. the sure its evolution as the concentrate on the basic echni)color singlet states died in more detail using chni)quark flavours in two- nformal windows have been g Wilson gauge action and e masses of 3-fermion bound ed with SU(2) gauge theory he fermion mass and lattice tudies of these theories have pretation of the results, and c cases [ ng-Mills theories with higher his theory is known to possess es nonperturbative knowledge quivalent to the adjoint repre- walking nning coupling (dashed line, below). ] that an ideal candidate for minimal -function becomes zero and the running β ]. 16 21 ]. For a related study of a QCD-like theory 20 – 3 – – 17 QCD−like IR fixed point -function of a theory with an infrared fixed point (dash-dott β β ]; in comparison to these studies we use considerably larger 18 . The schematic , 17 In this paper we consider the case of SU(2) gauge fields with tw Naturally, the most direct way to answer the question concer On the other hand, analytic semi-quantitative studies of co Figure 1 two-index symmetric representation, which,sentation. for SU(2), We is measure e coupling various in correlators order asof functions to of this obtain t theory. informationnon-improved on The Wilson the theory quarks. spectrum isrefs. This of discretised [ theory (t on has the been lattice studied usin on the coupling constantscale in is this changed. theory functionalSchrödinger would Although method, be this care to canwe is directly be leave needed mea this reliably in analysis measured the for on inter further t work. Here we will instead with fundamental representation fermions see [ able to single outlattice some methods. viable candidates Infermion which particular, representations should for it be non-supersymmetricwalking has stu Ya technicolor been theory suggested wouldindex be [ symmetric the one representation with ofbeen just SU(2) performed two or (te on SU(3). the lattice Initial already s [ volumes and different coupling constants,states. and we We also also measur makewith two direct flavours comparisons of fundamental with representationQCD-like results fermions. running obtain coupling T and chiral symmetry breaking. full quantitative detail, andwhich to the date reason can only is be that obtained this on the requir lattice. top), walking coupling (solid line, middle) and QCD-like ru feature an infrared stable fixed point at which the coupling freezes to a constant value. Beyond supersymmetri JHEP05(2009)025 ], ll 17 ), and viour is , κ 2. If this 0 limit is , whereas L 2 β / ∼ > 1 Q → L m the chiral scale β Q L m β ]. We remind here nning coupling and 18 ng constant the possible e parameters ( as the technifermion mass sical excitation spectrum. lt, because the comparison fermion mass is additively system: firstly, we map out eveals qualitatively similar and the masses of the two- ervables. In the conformal ng axial Ward indentity. It te. In this case all masses uss the implications this will is lowered. This includes the important to understand the his prevents us from reaching the chiral symmetry breaking ectrum of color singlet states lement the initial studies [ mewhere near vations [ er, the first order transition at mal invariance. Furthermore, 0. This behaviour is especially ce of a mass scale immediately cating that the phase structure the system has a rather strong on”, behaves as Q of the color singlet states in the 2 m → (asymptotic freedom), it may also . /g Q L 2 the behaviour is markedly different: β m = 4 and apparent restoration of it at large ∼ > →∞ L L L L β β β β – 4 – . What happens is that at large L . The line of first order transitions ends at around β L β 2. However, at , the inverse lattice spacing. Thus, very large volumes, sma ≈ we observe clear signs of chiral symmetry breaking: the mass /a L L β β behaviour does not correspond to continuum physics. it is possible to reach the zero fermion mass limit. This beha , and no phase transition is encountered when L Q L β β m prevents us from going to the chiral limit Q m 2, and at larger More precisely, we investigate the following aspects of the Secondly, we investigate the chiral behaviour of the system However, this interpretation must be taken with a grain of sa is decreased. At small ≈ . However, now we know for sure that the system has QCD-like ru Q L L masses of 3-fermion states, whichsmall exist in this model. Howev Even without directconformal knowledge behaviour of can the betheory, behaviour no characterised massive of bound through the states otherimplies can coupli breaking obs exist, of since the theshould existen conformal provide invariance. first Hence, hintsregarding the of sp the the possible possible conformalonset underlying or of confor chiral walking symmetry behaviour breaking it in these is theories. To comp features of the theory, the lattice phase diagram and the phy have a non-trivial fixed point at some finite value of that while the system has a continuum limit at we present high-precision results concerningSU(2) the gauge spectrum theory with twohave adjoint fermion on flavours the and possible disc conformal (or walking) behaviour. the phase diagram of the theory on the plane of the bare lattic prominent near the critical the masses of other states approach a constant value as m locate the line whererenormalised the with fermion Wilson mass fermions,turns vanishes. the out Because mass that the is at measured small usi inverse lattice coupling of the pseudo-Goldstone pseudoscalar 2-fermion state, “pi there we observe no clear signs of chiral symmetry breaking, fermion pseudoscalar andare vector proportional states to are almost degenera β observed to be independent ofpersists the lattice at volumes infinite used, indi volume. This also confirms earlier obser first order phase transition asvery the small fermion fermion mass masses is lowered. at T small approached. As will be discussedis below, consistent the with apparent lack the of existence of an infrared fixed point so with the theorybehaviour: with chiral fundamental symmetry representation breaking fermions at small r is the case, the small chiral symmetry breaking also at large β becomes much smaller than 1 JHEP05(2009)025 L β ns. (2.2) show all volume, ur. Furthermore, ivative is mation of the fermion lobal symmetry group ntation fermion theory ].) However, the results ng the existence of the r remains essentially the 19 n the small and large y breaking, i.e. coupled to he direct evaluation of the ysical low energy spectrum amics of two Dirac fermions In the second, “QCD-like” above alternatives, and our f. [ his case the apparent chiral , ifermions. ar conclusions were obtained s model has been proposed to c s inherent in the interpretation ], by extending these studies to u effect. d (2.1) em has a genuine infrared fixed 18 ume and resolution effect. In this / Q D d abc m ǫ b µ A u+ we recall the basics of the model as well / we present our results and interpretation D TC 2 u 3 ig + − – 5 – µν ac F δ µ µν ∂ F 1 2 = − a = u] . µ (1) gauge fields, three of these Goldstones will become the 4 L we conclude and outline the directions of our future work. D Y [ 4 U × and 24 . This pattern leads to the appearance of nine Goldstone boso where the chiral symmetry appears to become restored should i (2) 4 L L dd β 3. , 2 uu + the system also may become effectively deconfined due to too sm , h and high accuracy are required to observe the chiral behavio is the usual SU(2) field strength, and the gauge covariant der L β = 1 /a µν 1 F a As will be described in detail below, for the adjoint represe Due to the pseudoreality of the SU(2) representations, the g The paper is structured so that in section The continuum theory is defined by: Thus, our results lend support to the earlier claims favouri ≪ Q -function is required to unambiguosly resolve the issue. it is very difficultobservations to do unambiguously not distinguish yet betweenfor completely SU(3) the gauge resolve theory the with case. 2-index symmetric (Simil fermions in re where leading to effective restorationsymmetry of restoration is the a chiral small-volume symmetry. and too In large t at large m where of the above theorycondensate is SU(4), which breaks to SO(4) due to a for as of the lattice formulation we use, in section of the data and in section suggest that the firstpoint. alternative is In favoured, this i.e. case the syst there must be a genuine transition betwee much larger volumes. However, we alsoof stress the the ambiguitie results basedβ on the spectrum and chiral quantities. T 2 The model andThe lattice minimal model formulation to be studiedin here the consists adjoint of representation the of gaugebe dyn the (quasi)conformal SU(2) with gauge only theory. two Thi flavours of massless techn If the above theorythe is electroweak taken SU to drive the electroweak symmetr longitudinal components of the weak gauge bosons, and the ph infrared fixed point with associated conformal behaviour [ clear finite volume dependence.same for As volumes 10 shown below, the behaviou behaviour at the endpointalternative, the of chiral the symmetry restoration firstcase is order a the finite value transition vol of line. JHEP05(2009)025 2 bare (2.4) (2.3) (2.5) (2.6) (2.7) /g = 4 malised as L β . We note that T V . , for two (degenerate) = F any of the fields above. µ,µ arameters, S 1 − T − ates. In the adjoint theory tal fermions is the replace- x V sless) continuum theory, the C interactions. Here we are V ion fermions are considered. thermore expected to receive ) cannot avoid the inclusion of rform lattice simulations with , two quarks or a quark and an µ oup is /2). Finally there are possible , ructed in analogy to baryons of uge theory and do not consider ng the square root problem (for γ )) quark and technigluon. orm in the same representation, , x − ( µν f,y ; † µ x ψ U P b xy + (1 , S Tr ) F M 2 1 x S x,µ ( is related to the quark mass. Because the f,x V µ − ¯ + ) ψ κ U 1 µ G a γ S S x,y – 6 – X d = , µ<ν ; X (1 + x =u -matrices are real and X lat f L S V β µ ) = 2tr( = X ). The Wilson fermion action, x = ( κ F x 1 plaquette, written in terms of the SU(2) fundamental ( ) with the variables S G ab µ − µ x × S V ( U µ xy δ U )]. The parameter = bare xy q, M ]). 3 are the generators of the fundamental representation, nor am , 22 2 , . The elements of ab δ = 1 2 1 is the standard 1 [8 + 2( / a = µν a P S = 1 b S κ The lattice action is parametrised with two dimensionless p We will study the masses of several (techni)color neutral st On the lattice the action is a S representation link matrices and is expected to contain six Goldstone bosons and these are fur masses of the orderinterested of of the the electroweak strong scale couplingthe properties through, coupling of e.g. to the ET the SU(2) electroweak. ga the spectrum is muchBecause richer the than technigluons, quarks if and fundamentalwe antiquarks representat can all transf obtain gaugeThe singlet simplest states to by measure combining areantiquark. meson-like two The states three or quark consisting more color of ordinary neutral of states QCD, can e.g. be “proton” const (spin-1/2)exotic and states “delta” like (spin-3 the color neutral combination of a techni bare quark mass receives large additive corrections, and we Wilson fermion action breaks the chiral symmetry of the (mas where Dirac fermions in the adjoint representation of the gauge gr with the standard Wilson plaquette action where because the link matrices aretwo Dirac real, flavours it of would staggered be fermionsan possible without example, encounteri to see pe [ The only modification to thement usual of Wilson the action link for variables fundamen where Tr JHEP05(2009)025 ) ). a ( (2.8) (2.9) O 2.11 (2.11) (2.10) vanishes Q m -values around t 3, 1.7, 1.9, 2.2 and where . κ . , all our hadronic corre- i i = 1 0) i 0) o the time sliced averaged , L k mass is defined through , We also made an attempt standard conjugate gradi- 0) cessive statistical noise for β . We concentrate only on , corresponding to different ) correlators, but for these nditioned quark matrix and “proton” and spin-3/2 “∆”, , i 3 for the vector meson. We κ d(0 ark-gluon state; however, we γ u(0 , he simulations are performed f the simulation program and 5 5 2 k-antiquark states in eq. ( X d(0 γ γ , us, the lowest-order accuracy is 5 lar currents are as follows: example, the correlation function 0) γ , = 0)Γ 0) s. Except for the isosinglet channel, = 1 , y , , y X i ¯ d( y AP ) , ¯ d( i PP V ¯ d( ) ,t γ t V ) ∂ ,t x ,t = x 1 2 d( x ). In practice we use a few 5 ) discretisation errors, which could be elimi- d( L X u( γ a β X 5 0 ( – 7 – →∞ ( t γ γ c O ) ) )Γ κ = lim ,t ,t ,t x x Q x m ¯ ¯ d( d( ¯u( ) has h h h y y y , , , 2.5 the value of the hopping parameter x x x X X X ). L ) improved “clover” action. However, to fully implement β = 1) and axial vector (Γ a )= )= 2.8 )= we used 5 to 11 different values of ( t t t ( ( ( X O L X β PP AP G V V for the pseudoscalar and Γ 5 γ = X 2 for the fit in eq. ( / t The masses of colour singlet “hadrons” are estimated by fits t The correlation functions for three quark states, spin-1/2 The Wilson quark action ( L = nated by employing the improvement would substantially increaserequire the careful complexity o renormalisation ofjustified various in quantities. exploratory investigation Th with as the is hybrid Monte the Carloleapfrog case algorithm, integrator. here. using even-odd T preco Theent method. quark matrix is inverted using the correlation functions withflavour Coulomb non-diagonal gauge (isospin non-singlet) fixed operators.for wall For quark-antiquark sources states, “mesons,” is given by the flavour non-diagonal axial Ward identity the mass term in the lattice action. The physical (PCAC) quar where the pseudoscalar-pseudoscalar and axial-pseudosca are constructed in analogousto fashion measure to the standard correlation SU(3)did function QCD. not of succeed the inmeasuring colour the constructing singlet mass. a qu source operator without ex 3 Results The simulations were carried out with five different values of where Γ also measured scalar (Γ the statistical errors remain toostates large consisting for of reliable two result quarks are degenerate with the quar 2.5. For each value of t defines the critical hopping parameter Here and below welation use wall functions. sources For with each Coulomb gauge fixing in JHEP05(2009)025 . 9 . Q ase m = 1 √ =1.3 =1.7 =2.5 L L L L β β β ∝ β we show the 2 of the lattice coupling κ the phenomena of chiral igenvalues of the fermion roup and two flavours of more difficult the smaller ible. This becomes more (1). The most demanding SU(2)-fundamental depending strongly on the no such symmetry to begin d up to 200-300 trajectories ), defined as the line where ase diagram, we performed a ldstone bosons, whereas other ajectories were sufficient. For L ∼ O β ( ase diagram of the lattice theory. c δτ κ × s 0.1 0.15 0.2 0.25 for the theory with adjoint representation N 7 and 2.5. Some of the early results of

0.5 0.3 0.6 0.4 0.2 κ plaquette 02 for larger values of the quark mass and . = 1 lattices. Right panel: Same as left, but for the )), the mass of the pseudoscalars is β 4 L ( lattice. In the left panel of ﬁgure c β – 8 – . In addition, we made exploratory runs with was 0 4 4 =1.3 =1.7 =1.9 =2.2 =2.5 τ )) vanishes. If there is chiral symmetry breaking, L L L L L β β β β β κ < κ 2.8 5 near the zero quark mass limit, and also at . 0 ( and 32 4 > ]. Q 23 m 2 and 2 . 003 closer to the zero mass limit. The number of integration . κ = 2 L SU(2)-adjoint , but we did not use these volumes for spectroscopy. For each c β 4 as functions of the hopping parameter and 10 4 was chosen so that the trajectory length . Left panel: The plaquette expectation value at ﬁxed values 2 bare s /g 0.1 0.15 0.2 0.25

0.3 0.5 0.7 0.6 0.2 0.4 plaquette However, in practice the simulations become progressively As briefly discussed in the introduction, we investigate how = 4 L was decreased down tosteps 0 the number of hybridquark Monte mass. Carlo The trajectories evolution was timestep 100-1100, ∆ volumes 8 fundamental representation fermions. quark masses, with volumes 24 fermions. The measurements are done using 10 the quark mass (as defined in eq. ( β this study have appeared in [ Figure 2 simulations were at symmetry in the continuum emergeswith. from the lattice The which has standard way is to study the critical line Locating this critical line allows us to sketch a possiblethe ph quark mass is.matrix, This which is due finally to makesevere the in the appearance large inversion of volumes. of very Inset small the order of e to simulations matrix on better imposs a understand relatively the small ph 10 along this line the pseudoscalarhadrons “pions” remain become massive. massless If Go at the smallest quark massfor reached. thermalisation, At whereas these at pointscomparison larger we we quark discar masses also 10–30 performedfundamental tr representation simulations quarks, with using SU(2) gauge g JHEP05(2009)025 occurs for the at fixed =1.7 =2.5 2 L L L . In com- κ β β β 2 , measured in the case Q L m β 2, while at larger tal representation ∼ κ larger lattice volumes, nge of L,c e other hand, at larger we show β . We take this to be an SU(2)-fundamental 3 t panel of figure κ urns out to be very difficult t panel of figure discontinuity (or possibly no lue as a function of esentation quarks, respectively. ind of behaviour were also seen 2, at which point there possibly for different values of ∼ nel: Same as left but for fundamental κ as a function of the hopping parameter point, which may correspond to some kind Q L,c smaller than m inuum gauge theory, however. 0.1 0.12 0.14 0.16 0.18 0.2 1 9 we are not able to reach very small 0 . L

0.8 0.6 0.4 0.2 ]. These ﬁrst order transitions are lattice

-0.2 -0.4 1 < β β Q

a m 24 L 2 ∼ < β L – 9 – β . We note that a ﬁrst order phase transition at L has also been seen in standard SU(3) lattice QCD β =1.3 =1.7 =1.9 =2.2 =2.5 L L L L L L β β β β β β in SU(2) gauge theory with adjoint fermions. The results 2 bare . In left and right panels of ﬁgure 4 κ /g SU(2)-adjoint = 4 L β it is possible to do simulations at all values of ), as functions of the hopping parameter L 2.8 β . Left panel: The physical (PCAC) quark mass 0.12 0.15 0.18 0.21 0.24 0

0.3 0.9 0.6 1.2 Q or in some cases 32

This observation can be contrasted with the case of fundamen We measure the quark masses and the hadron spectroscopy from a m The first order transition line presumably ends at a critical 18 2 4 for the theory with adjoint representation quarks. Right pa fermions: we have performed the measurements for the same ra of continuum physics. This is unlikely to be the desired cont Figure 3 of fundamental fermions and the results are shown in the righ parison with adjoint fermions,discontinuity we at find all) considerably at smaller small values of 24 artifacts, not present in the continuum theory. κ small values of the lattice coupling adjoint representation quarks and forFor the the fundamental repr adjoint representation, at small using eq. ( quark masses, because the first order phase transition in lef with Wilson fermions; for a recent review, see [ lattice coupling representation fermions. results for the measurement of the plaquette expectation va show clear signs of a discontinuity for values of couplings we obtain a smoothto curve. make Even with simulations this at volumevalues it very of t close proximity to the jump. On th indication of a first order phase transition for is a critical point wherein the [ transition ends. Signs of this k JHEP05(2009)025 . , f 4 L ”) ak β π . In ∞ rst order = L we observe β ”) mass has L ρ , unless there β 8 at / aching very small →∞ L njectural; we cannot = 1 int theory in figure β ppen also in standard κ appears to terminate at nts of the mass spectrum d from smaller volumes; e discontinuity vanishes in issing first order transition behaviour for fundamental we show the pseudoscalar- alled Aoki phase, which is a ector meson (“ At this point the measured rm the behaviour established till require very short update ected to have similar features 7 ding to much higher values of alar is the Goldstone boson of cal point. We remind that the ector meson masses now appear his is done only for illustration; 3 . This is the expected behaviour limit is at ]. In this phase flavour symmetry Q 235, which is substantially beyond the . 27 m – = 0 √ 2, the location of the possible critical 25 κ . For strong bare coupling, i.e. small ∼ Q L m β -values corresponding even to negative quark 4 and extends towards ]. On the other hand, at larger / κ ]. We note that the lattice phase diagram of the – 10 – 24 18 = 1 κ values; the results for the vector mass are shown in 2, above which the behaviour appears to be regular L we show the pseudoscalar quark-antiquark meson (” leading to non-trivial infrared physics. β ∼ 5 L β 3 we are able to reach . 22. This is presumably due to the large metastability of the fi L,c . β 0 , and very nearly degenerate. for the theory with adjoint fermions. The close degeneracy o = 1 is increased above Q ≈ ρ L κ m L β β ) is defined as the line where the quark mass vanishes. In the we /m L π β ( m c = 0 it has the value κ L β -region, where the first order transition prevents us from re . L L β β . However, as = 0 -limit is passed. The location of this critical point is co 6 Q -values which correspond to non-zero quark masses. With these results in mind, let us now consider the measureme We conjecture that the first order transition leads into so-c We are now in position to sketch the phase diagram for the adjo As an alternative way to plot these results, in figure However, note that at κ m 3 , possibly to infinity. However, the abruptness with which th L transition at large volume. location of the transition at as at in the presence of chiralthe symmetry broken breaking, symmetry. if Consistently the with pseudosc this behaviour the v our simulations may suggest thetheory existence is of asymptotically a free, genuine thus, criti the lattice continuum theory with fundamental SU(2) representationto quarks the is adjoint exp quarkat one, small with the possible exception of the m The critical line and parity are spontaneously broken.in the Our earlier observations studies confi at smaller volumes [ mass as a function of the PCAC quark mass vector mass ratio PCAC quark masses abruptlynot jump shown to on negative theSU(3) values plot). (measure lattice QCD This with Wilson behaviour quarks has [ been observed to ha and their implications. In figure the small- numerically exclude a very weakβ first order transition exten coupling limit a finite intercept for these same no particular problems in simulating with quark masses, we nevertheless extrapolatewe its do location. not (T needa it critical in point subsequent roughly analysis.) at The first order line strong lattice coupling artifact for Wilson fermions [ exists a critical point at finite masses. Naturally, simulations atstep very small in quark HMC massesrepresentation trajectories. s quarks. We observe qualitatively similar point, there is a qualitative change: the pseudoscalar and v linearly proportional to figure we find that the pseudoscalar mass is proportional to JHEP05(2009)025 is L . We β 5. ρ . 0 = 0. At ent values ≤ Q a m we show the Q 8 m 2. ), where 9 and . L ∼ 1 β ( L c ≤ β κ ritical point shown by open L β se to the value 1.5, as 2.5 =1.3 =1.7 =1.9 =2.2 =2.5 L L L L L β β β β β = infinity rate with the spin-1/2 state. er than β ecture, and it can be more complex is evident. In figure = 0 q L m β 2 2 (“proton”) and the vector meson L a / β Q m – 11 – > 0 q 1.5 ) -plane for the adjoint theory. Solid circles denote the m 9) critical hopping parameters Pseudoscalar mass SU(2)-adjoint . ,κ to the mass measurements at 1 L β Q Aoki phase ≤ 1st order m L √ β β = 0 ∝ 1 -0.3 0 0.3 0.6 0.9 1.2 1 3 0 2 ”) mass for the adjoint representation quark theory at differ

0.1 0.2

0.5 2.5 1.5

π 0.15 π 0.25 a m κ . The phase diagram on ( . Pseudoscalar (“ 2 there appears a 1st order phase transition, which ends at a c . The dotted line is a fit L ∼ < β L Figure 5 of Figure 4 the pseudoscalar and vector meson masses at large mass ratio of the three quark state with spin 1 measured (and extrapolated for square. The phasethan structure shown above here. the The critical Aoki phase line may is exist a also conj for values larg β can observe that the mass ratio becomes flatter, and quite clo increased. The spin-3/2 three-quark state is almost degene JHEP05(2009)025 ]. 28 . If the Q m not quite reach is evident. . This behaviour is L e adjoint representation Q β m sition at small to ine of first order transitions =1.3 =1.7 =1.9 =2.2 =2.5 =1.3 =1.7 =1.9 =2.2 =2.5 L L L L L L L L L L β β β β β β β β β β near an infrared fixed point [ region has no chiral symmetry breaking. (small bare coupling) all measured masses a a L Q Q L β m m β SU(2)-adjoint ρ /m – 12 – π m Vector mass SU(2)-adjoint Vector mass ”) mass for the adjoint representation quark theory. ρ -0.3 0 0.3 0.6 0.9 1.2 0 0.3 0.6 0.9 1.2 1 1 3 0 2

1.1

0.5 0.8 0.7

0.6 0.9 0.4

. The near degeneracy of the masses at large

1.5 0.5 2.5

ρ π

/m m

ρ L a m β . Vector meson (“ we observe chiral symmetry breaking mass pattern, but we can L β . The ratio of the pseudoscalar and vector meson masses for th Figure 6 How to interpret these results? At large the zero quark mass limit because of the first order phase tran Figure 7 quarks at fixed values of At small theory has an exact IR fixed point, the large This region presumably ends at the critical point where the l in the adjoint quark theory are approximately proportional certainly compatible with a scale invariant behaviour at or JHEP05(2009)025 we show the ratio 9 iscussed in the intro- heory with fundamental ton”) and the vector meson figure =1.7 =2.5 =1.3 =1.7 =1.9 =2.2 =2.5 the theory with fundamental L L L L L L L ee and running and the theory β β β β β β β ttice artifact not present in the a a Q Q m m SU(2)-adjoint ρ SU(2)-fundamental ρ /m – 13 – P /m m π m but for fundamental representation quarks. 7 0 0.3 0.6 0.9 1.2 0 0.3 0.6 0.9 1.2 1 1 2

1.1

0.8 0.7 0.5 0.9 0.6

0.4 1.8

1.6 1.4 1.2

ρ π ρ /m m /m m . Same as figure . Figure 9 ρ /m . The ratio of the mass of the spin-1/2 three quark state (“pro P m However, this interpretation should be taken with care: as d mass, Figure 8 of the pseudoscalar and the vector meson masses in the SU(2) t start. In this case the first order transition is probably a la continuum theory. duction, qualitatively similar behaviourfermions where may the arise coupling evenis is in QCD-like, in asymptotically the fr chiral symmetry breaking phase. For comparison, in JHEP05(2009)025 , L we β L , and β ssess it /a . However, Q are required. m a Q the lattice spacing m is sufficiently large et: at small L L β sentation quarks is in β supports the view that ent evidence in support smaller than the hadron ble volumes. We discuss s at arbitrarily large 4 uch smaller than 1 lues of ke unparticle physics. We ever, there is an important resentation quarks, and it L ith care. By comparing to etry breaking on the weak al at small t behaviour. Naturally, our lution of the measurements heory and discussed its im- ion of the running coupling iour would be useful in the corresponding results in the joint representation case in e aim of the study has been re auge theory with two fermion lso known as walking techni- ample of (infrared) conformal at large s in the fundamental represen- y may be qualitatively different, ysical size of the lattice may be- 2, and its stability as the volume , we have analysed the spectrum symmetry breaking even at large ≈ 2 lattices we observe no clear signs . Finally, if . L 9 β = 2 9 we already see the first order phase . L β = 1 L , on β 4 where the apparent chiral symmetry restoration – 14 – L –32 β 4 can be seen in figure a Q m ) side of the critical point, the continuum theory does not po L β . In this case the reason for this behaviour is easy to interpr 7 Thus, the behaviour in SU(2) gauge theory with adjoint repre ; it may just evade detection within our accuracy and accessi becomes so small that the chiral symmetry breaking scale is m L size). This leads to behaviour which appears almost conform and the lattice size iscome kept so constant small in lattice that units, the the system ph becomes effectively deconfined ( in this case we know that the chiral symmetry breaking appear figure fermions. The result is qualitatively very similar to the ad the implications of our results in more detail below. 4 Conclusions and outlook We have investigated the strong coupling dynamicsflavours of in SU(2) the g adjoint representationto determine of whether the this gauge theoryor group. could provide almost Th a conformal concrete theory. ex application to Phenomenologically dynamical such electroweak behav color, symmetry or breaking, some a otherhave beyond determined Standard the model schematic phenomenologyplications phase li for diagram continuum of physics the andcase lattice also of fundamental t compared representation with fermions. the of Furthermore this theory. Evenof though conformal the behaviour, spectrum thesesimilar may arguments measurements be need carried argued out to for to be two pres flavours treated of w fermion observe the standard chiral symmetry breaking pattern, but provided that the latticeis volume sufficiently is high. large Thus, enough the and value of the reso it remains at infinitecoupling volume (large limit. If there is no chiral symm either. This would benumerical compatible results with cannot exclude the the infraredβ possibility fixed of poin chiral happens should depend on the system size, with faster evolut giving stronger volume dependence. major features quitemay similar be to tempting to the assume onedifference that which the with suggest physics to fundamental us is that similar rep namely the too. adjoint the fermion abruptness How theor of the change in behaviour at is changed. In our range of volumes 8 of chiral symmetry breaking, whereas at a transition. The stability of the critical point shown in figu in order to see the chirally broken mass pattern very small va This movement to small JHEP05(2009)025 , l of L the β ctric 2. Of L β : there nd thus ∼ 0. Also, 1 L is a lattice β -function of L β > β β al continuum limit mal. However, be- ng a small mass to , and there is spon- h from either of the breaking appears to uark mass effect, we d show clear volume alking” it interpolates aking are prohibitively → ∞ how the results presented ere are various options for ement of the L try breaking at ically free and for small ssibility of these behaviours mmetry breaking. Since the the scale dependence of the β inuum also for ree electric phase leading to , we conclude that the QCD- oints both in the infrared and Q ond order, or even non-analytic m 0 limit. However, at large → Q m – 15 – 0, our results show that the consistent continuum -function than the three sketched in figure → β the continuum limit is at L β above) is less likely than near or exactly infrared conforma 2, whereas just below this value we observe chirally broken I is large enough, at long distances the theory flows into the IR 0, corresponding to the dashed-dotted line on the right side ∼ > , there is also a further option corresponding to the free ele L while this belongs to the same universality class as case I (a L > 1 β . The observed chiral symmetry breaking at small β ) if g ( β → ∞ -function at the IR fixed point. β L β IR fixed point: is still at fixed point and there is no chiral symmetry breaking. The form Walking coupling: artifact not connected to continuum physics. lattice volumes required tolarge. observe the At chiral toocause symmetry small the bre volumes chiral the symmetry theorycan restoration may expect is look that a almost finitedependence, at confor which volume least we and some did q not of observe the in chiral this study. observables shoul has chiral symmetry breaking), dependingbetween on cases the I degree and ofcases II. “w above. Thus, it We can canthe obtain quarks, be this for very example. behaviour hard from to case distinguis II by addi QCD-like running coupling: taneous chiral symmetry breaking in the If we allow for Thus, in the light of the obervation that the chiral symmetry Nevertheless, even without such measurement we can discuss I) II) III) the fixed point in figure vanish immediately as yet another logical possibility wouldbehaviour be of a the single zero of sec like running coupling (case mass pattern and a first order phase transition at small evolution of the coupling. the theory flows into the IRcontinuum fixed limit point in and there this is case no is chiral at sy limit does not exist due to the observed onset of chiral symme this theory. here can be contrastedasymptotically with free the coupling: evolution of the coupling. Th tation, we have shown that it is not possible to rule out the po being due to finitecoupling volume constant artifacts. can Definitive only answer be about provided by a dedicated measur phase of the theory. In this domain the theory is not asymptot could be multiple zeros,ultraviolet. corresponding to In nontrivial such fixed a p case it would be possible to reach cont course, it is possibleeven that more chiral exotic symmetry scenarios is for broken the in the f JHEP05(2009)025 ]. , ]. ] , SPIRES , , SPIRES ]. [ ] [ Salam theory (2007) 275 , f strongly coupled SPIRES [ by US Department of s and comments. JT nd the excitation spec- 4371. AH acknowledges B 650 tions were performed at hep-ph/0703260 [ ion’s public lattice gauge presentation quarks. Our ]. ulich supercomputer center ects, the measurement of the eory in the continuum. How- hep-ph/0203079 ]. , e evolution of the coupling con- arXiv:0804.0182 , ces and add more solid evidence gredient for direct measurements (1981) 1441 s SPIRES ]. Phys. Lett. ] [ , SPIRES D 24 ] [ (2007) 221601 (2004) 553] [ 98 SPIRES Scale invariant technicolor model and a [ 390 Collider signals of unparticle physics – 16 – Phys. Rev. ]. ]; ]. , arXiv:0704.2588 [ (1986) 1335 arXiv:0810.2686 [ SPIRES 56 [ SPIRES Phys. Rev. Lett. SPIRES [ Strong dynamics and electroweak symmetry breaking [ Erratum ibid. , [ ]; Dynamical breaking of weak interaction symmetries Unparticle & Higgs as composites -function in the future. β (2007) 051803 SPIRES ] [ (1980) 125 (1979) 2619 (2003) 235 (2009) 015016 (1979) 1277 99 Phys. Rev. Lett. Implications of dynamical symmetry breaking: an addendum 381 Dynamics of spontaneous symmetry breaking in the Weinberg- , B 90 Dynamical stabilization of the Fermi scale: phase diagram o Raising the sideways scale D 20 ]. D 79 D 19 Another odd thing about unparticle physics Unparticle physics 29 ]. Phys. Rept. Phys. Rev. Phys. Lett. Phys. Rev. Lett. SPIRES arXiv:0704.2457 K. Cheung, W.-Y. Keung and T.-C. Yuan, Phys. Rev. Phys. Rev. [ [ L. Susskind, theories for (minimal) walking technicolor and unparticle technidilaton To summarise, we have determined the lattice phase diagram a [1] F. Sannino, [3] H. Georgi, [5] S. Weinberg, [9] K. Yamawaki, M. Bando and K.-I. Matumoto, [2] H. Georgi, [4] F. Sannino and R. Zwicky, [6] E. Eichten and K.D. Lane, [7] C.T. Hill and E.H. Simmons, [8] B. Holdom, References trum of SU(2) gaugeresults field support theory with earlier two studiesin flavours performed favor of on of adjoint the smaller re absence latti ever, of we chiral have emphasized symmetry that breaking due inspectrum to is this possible not th finite alone volume sufficient eff tostant. fully resolve Nevertheless, the these type results of provideof th an the important full in non-perturbative Acknowledgments We thank M. Antola, K. Kajantie and F. Sannino for discussion and KR acknowledge the supportpartial of support Academy by of the FinlandEnergy NSF grant grant 11 under under grant contract numberthe PHY-055375 DE-FG02-01ER41172. Center and for The Scientific(JSC). simula Computing Parts (CSC), of Finland, the andtheory at code code J¨ are [ derived from the MILC collaborat JHEP05(2009)025 ] ]; ] ] gauge SU(3) ]. flavors of ssless lattice Yang-mills , 2 SPIRES [ , SU(2) (2007) 034504 =4 , ]. with SU(3) SPIRES N ory ] [ arXiv:0803.1707 [ (2007) 085018 D 76 , The phase structure of SU(2) ]. hep-th/9509066 (1986) 957 SPIRES hep-lat/9812023 [ ] [ D 75 Spectrum of 57 -function in dhana, β -function SPIRES Probing technicolor theories ]. [ nen, (1998) 105017 Phys. Rev. β ]; ]; , r, Chiral hierarchies and the flavor (2008) 031502 ts: predictions for LHC arXiv:0712.0609 (1999) 449 PoS(LATTICE 2008)065 [ Phys. Rev. D 58 Nearly conformal electroweak sector , SPIRES ), ]; Phase diagram of SPIRES SPIRES D 78 ] [ N ]. ] [ ] [ ]. Phys. Rev. Lett. arXiv:0807.0792 B 550 [ ]. , SU( Light composite Higgs from higher (Proc. Suppl.) (1996) 1 [ Lattice study of the conformal window in SPIRES The ultraviolet finiteness of the [ Phys. Rev. Zero of the discrete arXiv:0805.2058 , SPIRES [ , SPIRES (2008) 171607 – 17 – Phys. Rev. [ Higher representations on the lattice: numerical SPIRES , [ 45BC ]. (2008) 009 Nucl. Phys. 100 flavors , ) Lectures on supersymmetric gauge theories and 11 hep-ph/0405209 hep-ph/0505059 arXiv:0711.3745 [ [ [ f ( Deconfinement and chiral symmetry restoration in an Walking in the Orientifold theory dynamics and symmetry breaking Supersymmetry inspired QCD (1982) 189 SPIRES N Minimal walking on the lattice [ ]. ]; ]. ]. (1983) 323 JHEP Nucl. Phys. , arXiv:0809.4890 , On the phase structure of vector-like gauge theories with ma , B 196 arXiv:0809.4888 SPIRES SPIRES B 123 , with adjoint fermions SPIRES SPIRES Phys. Rev. Lett. ] [ ] [ , (1977) 199 (2005) 051901 (2005) 055001 (2008) 065001 ] [ ] [ Charge renormalization in a supersymmetric Yang-Mills the SU(2) B 72 D 71 D 72 D 78 gauge theory with Nucl. Phys. ) ]. ]. ]. , Phys. Lett. N , SU( arXiv:0810.3722 SPIRES SPIRES hep-ph/0611341 arXiv:0705.1664 SPIRES hep-ph/9806472 with staggered fermions Theory theory with two fermions in the adjoint representation with chiral fermions [ simulations. [ L. Del Debbio, A. Patella and C. Pica, [ [ T. Appelquist, A. Ratnaweera, J. Terning and L.C.R. Wijewar [ Phys. Rev. Phys. Rev. D.D. Dietrich and F. Sannino, QCD-like theories gauge theory with adjoint[ fermions Phys. Lett. [ Phys. Rev. fermions electric-magnetic duality D.D. Dietrich, F. Sannino and K. Tuominen, representations versus electroweak precision measuremen dynamical adjoint quarks gauge theory with color sextet fermions an changing neutral current problem in technicolor [23] A. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuomi [22] F. Karsch and M. Lutgemeier, [20] Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroede [16] F. Sannino and K. Tuominen, [12] T. Banks and A. Zaks, [17] S. Catterall and F. Sannino, [18] S. Catterall, J. Giedt, F. Sannino and J. Schneible, [19] Y. Shamir, B. Svetitsky and T. DeGrand, [21] T. Appelquist, G.T. Fleming and E.T. Neil, [11] T.A. Ryttov and F. Sannino, [13] D.R.T. Jones, [14] L. Brink, O. Lindgren and B.E.W. Nilsson, [15] K.A. Intriligator and N. Seiberg, [10] T.W. Appelquist, D. Karabali and L.C.R. Wijewardhana, JHEP05(2009)025 , ]. ]. SPIRES ] [ ]. SPIRES ]. CD with Wilson fermion [ Higher representations on SPIRES ] [ nino, SPIRES [ arXiv:0802.0891 [ arXiv:0810.5634 , . hep-lat/9804028 (2008) 007 [ Spontaneous flavor and parity breaking with Wilson arXiv:0811.0616 – 18 – ]. , 06 JHEP SPIRES [ , (1998) 074501 D 58 (1987) 140 B 190 Conformal chiral dynamics Lattice QCD: a critical status report Phys. Rev. Numerical evidence for a parity violating phase in lattice Q , http://physics.utah.edu/˜detar/milc.html Phys. Lett. fermions the lattice: perturbative studies [25] S. Aoki, [26] S.R. Sharpe and J. Singleton, Robert L., [27] L. Del Debbio, M.T. Frandsen, H. Panagopoulos and F.[28] San F. Sannino, [29] See [24] K. Jansen,