PoS(Lattice 2012)023 ) 2 ( SU sentation http://pos.sissa.it/ over in the chiral condensate the phase structure of a ver, if the gauge coupling is strong nsistent with the usual continuous ssover in the behavior of the chiral ent in a reduced staggered fermion invariant four fermion term. Pairs he generation of a dynamical mass ssary for generating the four fermion tice 2012, e Commons Attribution-NonCommercial-ShareAlike Licence. Dirac fermion flavors transforming in the fundamental repre f N † ∗ [email protected] [email protected] Speaker. Work supported in part by DOE grant DE-FG02-85ER40237 We report on Monte Carlo simulations focused on elucidating of the group and interacting through an additional chirally formulation of the theory with the Yukawa interactions nece condensate for strong four fermi coupling associated with t of physical flavors are implemented using the two tastes pres term preserving the usual shift symmetries. We observe a cro for the fermions. At weakphase gauge transition coupling seen this in the crossover pure is (ungauged) co NJL model. Howe enough to cause confinement weconsistent observe with a a much first more order rapid phase transition cross gauge theory containing ∗ † Copyright owned by the author(s) under the terms of the Creativ c

Simon Catterall Four fermion operators and the searchPhysics for BSM Physics Department, Syracuse University E-mail: Aarti Veernala Physics Department, Syracuse University E-mail: The 30 International Symposium on LatticeJune Field 24-29, Theory 2012 - Lat Cairns, Australia PoS(Lattice 2012)023 with a ractions dard Model Simon Catterall el arises as an del building, as solution to these lly invariant four tion to check the le due to the fine se the addition of to the Schwinger- . N. It is widely be- mions are charged scalar top–anti-top om strongly bound e phase diagrams of ries which, for zero eories. eaking may be made exhibit very different sults that indicate that critical behavior in the ttice construction. The [5, 6, 7, 8]. In the latter ntrated on the four flavor y between the symmetric o four fermi coupling and re NJL, we shall argue that n gauge theories in general. en in the continuum that the els depends on the approximation that 2 [9, 10]. The evidence for this behavior < µ γ . Understanding the effects of the four fermion < ) 2 2 ( SU corresponding to the appearance of a line of new fixed points associated 1 consistent with the presence of any new fixed points in the theory. not In the work reported here and described in detail in [11] we have conce Our motivation in this paper is to study how the inclusion of such four fermion inte The focus of the current work is to explore the phase diagram when fer Notice that the appearance of a true phase transition in the gauged NJL mod Elucidating the nature of the electroweak symmetry breaking sector of the Stan However, to obtain fermion masses in these scenarios requires additional mo 1 term in this theoryfour can then fermi serve coupling, as lie aa near four benchmark or fermion for inside term future the will studiesarbitrarily conformal small break of window. by conformal tuning theo invariance the In but four the fermisuch in coupling. conformal latter principle It or ca that is walking entirely br theories possible infeatures that th than the those presence seen of for four a fermi confining terms gauge may theory. theory corresponding to two copies offour the flavor basic theory Dirac is doublet expected usedis to in the free be la from chirally sign broken problems and for confining gauge at group zer may influence the phase structure and low energySpecifically behavior we of non-abelia have examined afermi model interaction with - both a gauge model interactions known and in the a literature chira as the gauged NJL model [9] derives from calculations utilizing theDyson ladder equations. approximation A in primary Landauvalidity goal of gauge of these conclusions the and current specificallygauged study to model search was for as to qualitatively compared new use to the latticethe pure simula phase NJL structure theory. of While the we gaugedour will NJL model results present is are re indeed different from pu (SM) is the main goallieved of that the the Large simplest Hadron scenariotuning Collider involving and currently a running single triviality at scalar problems Higgs CER whichproblems field arise can is in be untenab scalar found fieldeffective by field theories. assuming theory One describing that natural fermion-antifermion the the pairs. dynamics Higgs These of models sector are a in generically composite called the technicolor field th Standard arising Mod fr in extended technicolor models [1, 2,models 3, 4] four-fermion and models interactions of drive top-condensation bound the state formation which plays and the condensation role of of the Higgs a at low energies. Four fermion operators 1. Introduction mass anomalous dimension varying in the range 1 under a non-abelian gauge group.gauged NJL Indeed, model may arguments exhibit have differentand been critical broken giv phases behavior at the boundar we can neglect the running of the gauge coupling PoS(Lattice 2012)023 R ) the 2 ( (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.1) ) gauge h SU − × x ( L ) φ 2 h Simon Catterall ( which is to be ∑ ) x 1 SU 16 ( lism with the result . ermion doublet we 3 are the generators χ )] ) x x ... ( )= ( ] es the form x 1 2 µ ( αε ) i ξ . φ = ) e ) ψ field acting on the reduced x x a a ( ( , τ ∗ → µ ε a 5 ) ) γ τ U i x x ¯ ( ( )] allows us to show that the Pfaffian ψ µ x χ . ) ( φ ) 2 ε +( ( ρ 2 G , − ) ) + . The required pseudofermion weight for SU 1 ρ 2 x [ )+ ( ¯ , 1 2 ψψ x + µ ) ( ρ [( x φ γ ρ µ 2 doublets of gauged massless Dirac fermions symmetry ( f 2 / )+ η µρ Ψ ) χ + f N x δ 5 G ) 1 subgroup of the continuum axial flavor transfor- ( ) 2 x )[ γ 3 N x ( ( 1 µ µ ( ∗ µ gauge group and incorporating an − U ρ a U − → ξ ( ) U ψ )] + 2 , ) Ψ x ( ] x / ( A → → → ( discrete ε according to µν χ ) ) ) SU − x x x F + ) Ψ / ( ( ( the usual gauge coupling and ∂ x i 1 µ µ ( χ µν [ ( g φ µ F U 1 2 [ ψ U x Tr 4 ) are the usual staggered fermion phases, )= x 2 d ( x g ) 1 ( . The pseudoreal character of T x 2 depending on the gauge field Z 2 µ ( χ / ε f )) U = − symmetry corresponding to fermion number. More interestingly it is also N µ , U ) ∑ x ) ( S 1 and M = ( M ( ) ( S U x ( µ ξ flavour group. , ) ) x 2 ( ( µ η SU We have used the RHMC algorithm to simulate the lattice theory with a standard Wilson Clearly the theory is invariant under the Note that the fermion operator appearing in eqn. 2.2 is antisymmetric We will consider a model which consists of This action may be discretized using the (reduced) staggered fermion forma 2 flavors is then Pf f chirally invariant four fermi interaction. The action for a single doublet tak mations which act on the matrix field in the fundamental representation of an action being employed for theobtain gauge a Pfaffian fields. Pf Upon integration over the basic f where These shift symmetries correspond to a N Notice that no single site mass term is allowed in this model. 3. Numerical results average of the scalar fieldstaggered over fermions the takes the hypercube form: [12, 13] and the gauge where G is the four-fermi coupling, Four fermion operators 2. Details of the model of the invariant under certain shift symmetries given by interpreted as the PoS(Lattice 2012)023 (3.1) 16 and × 3 iety of gauge Simon Catterall and 8 phase transition. hich is computed varies holding the 4 r strong gauge cou- β , 8 ur fermi coupling at g artifacts and so we 8 the plaquette drops se transitions in this ) . 4 x 1 ious simulations using ( e as , 6 multiples of four flavors µ < 4 ξ ) β x ymmetries discussed earlier. ( ε . The results in this paper are f
8 N ) µ 8. For − . The crossover coupling is volume e ) = β M − L † x 1 for four flavours ( . M 0 χ ( ) = µ e G − . We have set the fermion mass to zero in all of at 0. To determine where the pure gauge theory is x . β ( β 4 † µ 10 U < 2 g )+ / µ 4 for lattices of size . 4 e 2 + ≡ ∼ x 1. This is shown in figure 1. We see a strong crossover between β ( c . 0 for a range of different lattice sizes. The chiral condensate is . 10. This is consistent with earlier work using sixteen flavors of β 0 χ Polyakov loop vs 2 < ) to a deconfined regime at large x = = 8 ( . = β µ β G β U 4. We have utilized a variety of lattice sizes: 4 ) = x Figure 1: 9 for f ( . 0 χ N lattices. Notice the rather smooth transition between symmetric and broken 4 ∼ G on 8 4. Here a very sharp transition can be seen reminiscent of a first order . β 2 ≤ β This behavior should be contrasted with the behavior of the condensate fo In Figure 2 we show a plot of the absolute value of the condensate at a var One of the primary observables used in this study is the chiral condensate w pling phases around naive fermion reported inregion [14] of which parameter identified space. aconventional staggered line It quarks of [15]. also second agrees order with pha the behavior seen in prev couplings a range of gauge couplings 1 four fermi coupling fixed at our work so that our lattice action possesses the series of exactfrom chiral the s gauge invariant one link mass operator dependent and takes the value of In Figure 3 we highlightthe this single gauge by coupling showing a plot of the condensate versus fo devoted to the case strongly coupled and confining we have examined the average Polyakov lin Four fermion operators is purely real and so we arecorresponding guaranteed to to a have pseudofermion no sign operator problem of if the we form use a confining regime for small below 0.5 which we take ashave indicative confined of our the simulations presence to of larger values strong of lattice spacin PoS(Lattice 2012)023 Simon Catterall el is no longer me dependent it by adjusting the ritical value. This 4. 4. = f t of a first order phase ory. However, it jumps ue that this is to be ex- = N f ependence on the volume N with 4 and 8 4 lattice with 4 , 6 4 for the 8 β 5 0 for lattices 4 . 2 = for varying β G at G vs vs χχ χχ Figure 2: Figure 3: What seems clear is that the second order transition seen in the pure NJL mod now non-zero even for small fourconsistent with fermi spontaneous coupling chiral and symmetry shows breaking noabruptly in strong the to pure d much gauge the largercrossover values or when transition the is four markedlytransition. fermi discontinuous coupling in Indeed, character exceeds while - some the reminiscen appears c position to of get the sharper phase with increasing transition volume. is only weakly volu Four fermion operators present when the gauge couplingpected is – strong. in In the the gauged next model section we one will can arg no longer send the fermion mass to zero PoS(Lattice 2012)023 with chiral ) 2 y. How- ( el for four SU oupling, and . This is qual- Simon Catterall for ccompanied by 0 this transition G ndensate will re- continuous = g QCD interactions to maintain an exact G transition which exists g gauge coupling and hiral symmetry break- eration for sufficiently s phase transition in the gauge group. We have h to cause confinement. iral symmetry by gauge rder parameter for such probably best thought of ) ling. Notice that this type comes strongly enhanced rmion top quark operator. 2 l fluctuations in the chiral ( in the chiral condensate and we attribute it to the fact that ew mechanism for dynamical SU ate even for we find evidence for a continuous phase . Notice however that we see no sign that ∞ G → controlled by the four fermi coupling and cannot in the confining regime at small β 6 β not ) survives at weak gauge coupling. However our results indicate that ∞ = β 1 corresponding to the expected spontaneous breaking of chiral symmetr ∼ G . G shows that the chiral symmetry of the theory is already broken as expected c 4 flavors. This breaking of chiral symmetry due to the gauge interactions is a G We have examined the model for a variety of values for lattices size, gauge c Thus our results are consistent with the idea that the second order phase In this paper we have conducted numerical simulations of the gauged NJL mod The fact that we find the condensate non-zero and constant for stron = < f transition for flavors of Dirac fermion in the fundamental representation of the employed a reduced staggered fermion discretizationsubgroup scheme of which the allows continuum us chiral symmetries. ever, for gauge couplings that generate a non-zero chiral condens itatively different from the behavior ofthe regular reduced staggered formalism quarks and does not allow for a single site mass term or an exact or crossover appears muchcondensate. sharper and there is no evidence of critica 4. Summary symmetry. Thus the spontaneous breaking of theinteractions residual will discrete not lattice ch be signaledceive contributions by only a from light massive states. Goldstone Theas pion transition a and we crossover observe phenomenon the is corresponding measured tomass co the generation sudden due onset to of the a strong n four fermi interactions. Four fermion operators four fermi coupling since iting. receives a Indeed contribution the from measured gaugea one mediated transition link c since chiral we condensate observe operatorthis it condensate is to depends not be on non-zero an the for o gauge all coupling four fermi interaction strength. In the NJL limit any continuous transition ends if the gauge coupling becomes strong enoug In this case we dolarge however four see fermi evidence coupling of associated additional witha dynamical an possible observed mass first rapid gen order crossover phase transition. in the pure NJL theory ( system as we increase the fourfor large fermi coupling - rather the condensate be to sent to zero by tuning the four fermi coupling - there can be no continuou the generation of a non-zero fermionof mass even scenario for is small actually four fermi true coup are of already top expected quark to condensate break modelsThe chiral in magnitude of which symmetry this the independent residual stron fermion of mass a is four fe N G PoS(Lattice 2012)023 , Nuc. , vol. 90, Phys.Rev. , vol. D41, Simon Catterall lab. Phys.Lett.B Phys.Rev. , vol. 155, no. 1, pp. 237 – , vol. B221, p. 177, 1989. 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