Contribution to proceedings of the 33rd International Symposium on Lattice Field Theory (Lattice 2015), 14-18 July 2015, Kobe, JapanWe present a lattice study of a four fermion theory, known as Nambu Jona-Lasinio (NJL) theory, via Wilson fermions. Four fermion interactions naturally occur in several extensions of the Standard Model as a low energy parameterisation of a more fundamental theory. In models of dynamical electroweak symmetry breaking these operators, at an effective level, are used to endow the Standard Model fermions with masses. Furthermore these operators, when sufficiently strong, can drastically modify the fundamental composite dynamics by, for example, turning a strongly coupled infrared conformal theory into a (near) conformal one with desirable features for model building. As first step, we study spontaneous chiral symmetry breaking for the lattice version of the NJL model

Drach, V

Hietanen, A

Pica, C

Rantaharju, J

Sannino, F

arXiv

Contribution to proceedings of the 33rd International Symposium on
  Lattice Field Theory (Lattice 2015), 14-18 July 2015, Kobe, JapanWe present a lattice study of a four fermion theory, known as Nambu Jona-Lasinio (NJL) theory, via Wilson fermions. Four fermion interactions naturally occur in several extensions of the Standard Model as a low energy parameterisation of a more fundamental theory. In models of dynamical electroweak symmetry breaking these operators, at an effective level, are used to endow the Standard Model fermions with masses. Furthermore these operators, when sufficiently strong, can drastically modify the fundamental composite dynamics by, for example, turning a strongly coupled infrared conformal theory into a (near) conformal one with desirable features for model building. As first step, we study spontaneous chiral symmetry breaking for the lattice version of the NJL model

Drach, V::0000-0001-8948-0220

Plymouth Electronic Archive and Research Library

arXiv:1511.03899v1  [hep-lat]  12 Nov 2015Wilson Fermions with Four Fermion InteractionsJarno Rantaharju∗CP3 -Origins, IFK & IMADA, University of Southern DenmarkE-mail: rantaharju@cp3.dias.sdu.dkVincent DrachCP3 -Origins, IFK & IMADA, University of Southern DenmarkE-mail: drach@cp3.dias.sdu.dkAri HietanenCP3 -Origins, IFK & IMADA, University of Southern DenmarkE-mail: hietanen@cp3.dias.sdu.dkClaudio PicaCP3 -Origins, IFK & IMADA, University of Southern DenmarkE-mail: pica@cp3.dias.sdu.dkFrancesco SanninoCP3 -Origins and the Danish IAS, University of Southern DenmarkE-mail: sannino@cp3.dias.sdu.dkWe present a lattice study of a four fermion theory, known as Nambu Jona-Lasinio (NJL) the-ory, via Wilson fermions. Four fermion interactions naturally occur in several extensions of theStandard Model as a low energy parameterisation of a more fundamental theory. In models of dy-namical electroweak symmetry breaking these operators, at an effective level, are used to endowthe Standard Model fermions with masses. Furthermore these operators, when sufficiently strong,can drastically modify the fundamental composite dynamics by, for example, turning a stronglycoupled infrared conformal theory into a (near) conformal one with desirable features for modelbuilding. As first step, we study spontaneous chiral symmetry breaking for the lattice version ofthe NJL model.Preprint: CP3-Origins-2015-045 DNRF90, DIAS-2015-45The 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/Four Fermion Interactions with Wilson Fermions Jarno Rantaharju1. IntroductionDespite the apparent Standard Model-like nature of the Higgs sector, it might still conceal,in plain sight, a more fundamental composite nature [1]. In fact, it has been recently shown thatgauge-Yukawa theories, similar to the Standard Model, even if perturbative for some energy rangecan still abide compositeness conditions [1,2]. In this case the Higgs-like state at higher energies isnot a propagating degree of freedom and effective four-fermion interactions appear, and the theorybecomes a gauged NJL theory.Traditional models of fundamental composite dynamics, of either Technicolor or compositeGoldstone Higgs, make also extensive use of four-fermion interactions. This is clear from a recentattempt [3] to generate the top mass in a composite theory able to bridge between the Technicolorand Composite Goldstone Higgs limit.For these kind of models three types of four-fermion operators generally occur:Leff =aΛ2UV( ¯ΨSMΨSM)2 +bΛ2UV¯ΨSMΨSM ¯ΨTCΨTC +cΛ2UV( ¯ΨTCΨTC)2.The first term, involving only Standard Model fermions, can be suppressed by the large cutoff scaleΛ2UV . The other two may be enhanced by the dynamics of the Technicolor sector. The second termprovides the Standard Model fermion masses and the third one adds to the Technicolor dynamics.According to Holdom’s insight [4], models of walking dynamics, possessing large mass anoma-lous dimensions, can enhance the SM fermion mass-term operator dynamically. It was later recog-nised [5] that an ideal way to achieve walking dynamics (iWalk) is to ask the third operator (i.e.the one containing only the technifermions) to induce chiral symmetry breaking when added to anotherwise conformal Technicolor dynamics [5, 6]. Our ultimate goal is therefore to investigate thenonperturbative dynamics of the gauged NJL model. We investigate on the lattice the un-gaugedversion, i.e. the pure Nambu Jona-Lasinio (NJL) model, with Wilson fermions and furthermore weretain only the last four-fermion operator. This is the stepping stone for the gauged version. A sim-ilar model has been previously studied with the goal of understanding the phase structure of Wilsonfermions [7–9]. Models with staggered fermions have been studied in previous works [10–14] andchiral symmetry breaking has been observed.It is worth noting that the model is only effective. It is not renormalizable and does not havea continuum limit. Thus different lattice discretizations and the continuum are essentially separatemodels, although they share fundamental properties. There is, however, a scaling region wherephysical results depend only weakly on the lattice spacing a.2. The ModelWe study the NJL model with 2 flavors of fermions possessing 2 extra color degrees of free-dom. In this case one can construct an NJL action that preserves the full chiral symmetry. Unfor-tunately when representing the fermion fields with pseudofermions, the action must be renderedquadratic using auxiliary fields and the fermion determinant becomes complex1. We will therefore1It is possible to render the fermion determinant positive if the number of colors is even and there is no gaugeinteraction [7–9]. The remedy is not applicable here since we plan to generalize the study to a gauged model.2Four Fermion Interactions with Wilson Fermions Jarno Rantaharjustudy a model that preserves only a U(1)×U(1) subgroup of the original SU(2)×SU(2) symmetrygroup.The model is defined by the continuum Lagrangian˜L = ¯Ψ/∂Ψ+σ ¯ΨΨ+pi ¯Ψiγ5τ3Ψ+σ 2 +pi24γ2 (2.1)〈σ〉= 2γ2〈¯ΨΨ〉〈pi〉= 2γ2〈¯Ψiγ5τ3Ψ〉. (2.2)Using the Wilson discretization of the pseudofermion representation we have:L = χ†(M†M)−1χ + σ2 +pi24γ2 (2.3)M = /∂W +σ +piiγ5 (2.4)The Wilson term breaks chiral symmetry and introduces additional explicit chiral symmetrybreaking terms in the Lagrangian. In a renormalizable model the only necessary correction, a massterm, arises as a result of a divergence in the Wilson term [15]. It is manifestly of first order ina. All other terms vanish at small a. In the model under question new terms may also arise withpowers of γ and diverge at small a.Chiral symmetry is restored by a correction to the action such that the axial current(A3µ)isconserved:∂µ〈A3µ(x)O〉= 0The correction may be found, for example, via the PCAC relation. With the action 2.3 we have∂µ〈A3µ(x)O〉=〈aX3(x)O〉, (2.5)where aX3 is the variation of the Wilson term. This term includes the divergent contribution that iscancelled by the fermion mass term. It is instructive write down it’s renormalisation, expanding tothe first few lowest order operators:aX3(x) = a ¯X3(x)+cm (γ/a)aP3(x)+ (ZA (γ/a)−1)∂µA3µ(x) (2.6)+acA,1 (γ/a)∂µ∂µP3(x)+a2cA,2 (γ/a)∂(µ∂ν∂ν)A3µ(x)+a2cγ (γ/a)S0(x)P3(x)+ · · ·where a ¯X3(x) includes only vanishing contributions to the PCAC relation, P3 is the pseudoscalardensity and S0 is the scalar density. The second term on the right hand side has to be cancelled byadding a mass term to the action and the third one is a multiplicative correction to the axial current.The two terms on the second line can be seen as corrections to the axial current. The last termincludes the first new contribution to the action with an effect on chiral symmetry. It is related tochoosing the quadratic coupling of the auxiliary fields to be different.Each coefficient in eq. 2.6 can be expanded in γ to reveal contributions divergent in a. This is anatural consequence of the nonrenormalizable nature of the model. Any number of these terms maybe taken into account in the action and in the renormalization of lattice operators. For simplicity3Four Fermion Interactions with Wilson Fermions Jarno Rantaharjuwe study a model with only the mass correction. We include the cγ term only to study its effect.The full Lagrangian is thusL = χ†(M†M)−1χ + σ24γ2 +pi24γ2 +δγ(2.7)M = /∂W +m0 +σ +piiγ5. (2.8)In the following we refer to the physical mass as m=m0+cm/a and the correction as ¯δ = δγ +a2cγ .The restoration of chiral symmetry implies that all the terms on the right hand side in equation 2.5cancel.At a given γ , when a is too small, the divergent contributions may become relevant. On theother hand, when a is too large, the corrections arising with positive order of a will be significant.Only the region in between will provide physically interesting results.An additional complication arises because of the auxiliary field pi(x). There are significantdisconnected contributions in the mesonic triplet channels. The disconnected part increases with γ ,and is relevant for the pseudoscalar correlators.The meson masses are measured from correlators of the typeCΓ(t0, t) =〈∑x¯Ψ(x, t0)Γτ3Ψ(x, t0)∑y¯Ψ(y, t)Γτ3Ψ(y, t)〉(2.9)limt−t0→∞CΓ(t0, t) = AΓemΓ(t0−t) (2.10)For the vector meson mass (mρ ) it is sufficient to measure the correlator with Γ = γk and fitting itto the exponential at large t − t0. This correlator does not suffer significantly from disconnectedcontributions. We measure the pseudoscalar mass (mpi ) by from the correlator with Γ = γ0γ5 and,when possible, Γ = γ5 using the generalized eigenvalue method. We also measure the pseudoscalarmass (mpi2 ) using the correlatorCpi(t0, t) =〈∑xpi(x, t0)pi(y, t)〉(2.11)This measurement is useful when the disconnected contribution is large. The correlator is noisy,but can be evaluated without inverting the fermion matrix.3. Numerical ResultsFirst we study the correction ¯δ and its effect on chiral symmetry and the fermion mass. To thisend we simulate the model with a small coupling γ = 0.2a and several values of δγ . In all cases thelattice size is 16×83. When ¯δ is nonzero, the chiral symmetry is not restored at zero fermion massm. We can therefore study the effect of the correction by separately finding the line where m = 0and the chirally symmetric line.To find the chirally symmetric line we measure the pion mass mpi and the divergence of theaxial current normalized by the pseudoscalar densitym˜ =∑x,y ∂x0〈A3µ(x0,x)A3µ(0,y)〉∑x,y〈P3(x0,x)A3µ(0,y)〉 . (3.1)4Four Fermion Interactions with Wilson Fermions Jarno RantaharjuThe restoration of chiral symmetry implies m˜ = 0.To study the mass term we must define it without reference to the axial current. To determinea zero mass value we use mσ = m0 + 〈σ〉. At small coupling this coincides with the fermion mass.In general the expectation value 〈σ〉 is related to the chiral condensate (equation 2.2). At a finitelattice size there is always a small but nonzero chiral condensate, which changes sign when thefermion mass crosses zero. This is seen as a first order transition in mσ . A similar method was usedin meanfield theory in [16].-0.5 -0.4 -0.3 -0.2m0-0.2-0.100.10.2mσδγ=-0.1575δγ=-0.12δγ=0δγ=0.2-0.5 -0.4 -0.3 -0.2 -0.1 0m0-0.2-0.100.10.2δγ=-0.1575δγ=-0.12δγ=0δγ=0.2m˜-0.5 -0.4 -0.3 -0.2 -0.1 0m000.10.20.30.40.5mpiδγ=-0.1575δγ=-0.12δγ=0δγ=0.2-0.38 -0.36 -0.34 -0.32 -0.3 -0.28m0-0.2-0.100.10.20.3δ γm=0mpi=0mtilde=0Figure 1: Fermion and meson masses at small coupling.We have γ = 0.2a for several values of δγ . Upperleft: the bare propagator fermion mass. Upper right: m˜. Lower left: the pion mass. Lower right: the pointsin the δγ and m0 plane where the each mass crosses zero.The masses are shown in figure 1. We observe a jump in the mσ across zero when δγ ≤ 0. Atδγ > 0 we do not observe a jump. In each case we find mσ = 0 with a linear extrapolation usingpoints where |mσ |> 0.05. We find the chirally symmetric point with linear extrapolations of m˜ andmpi using points where mpi > 0.1. The resulting values are shown in figure 2.When δγ = 0 the jump in mσ and the restoration of chiral symmetry happen at the same baremass. This confirms the expectation that a2cγ is negligible at small γ . At δγ < 0 chiral symmetryis restored at positive mσ . At δγ > 0 we do not observe a jump when mσ crosses zero, but chiralsymmetry is restored at a negatife value of negative mσ . In all cases a nonzero value for the5Four Fermion Interactions with Wilson Fermions Jarno Rantaharjucorrection produces a noticeable but relatively small difference between the two critical lines.Having established the restoration of chiral symmetry at small coupling, we study its sponta-neous breaking at large coupling. We run simulations at γ = 0.65a and 0.6a and δγ = 0. The latticesize is again 16×83. We measure the pseudoscalar meson masses mpi and mpi2 and the vector mesonmass mρ . Chiral symmetry is restored when mpi = mpi2 = 0 and broken when mρ is different frommpi and mpi2 .-4 -3 -2 -1 0m000.511.522.53mpimρmpi2<pi>-3 -2.5 -2 -1.5 -1 -0.5 0m000.511.522.53mρmpimpi2<pi>Figure 2: The pseudoscalar and vector meson masses large coupling 0.65a (left) and 0.6a (right). We alsoshow the expectation value 〈pi〉, which serves as the order parameter for the parity broken phase.We show the masses of the pseudoscalar and the vector mesons in figure 2. We also showthe expectation value 〈pi〉, the order parameter for the parity broken phase. At large mass thedisconnected contributions to the pseudoscalar correlators become problematic. It is clear, however,that at small mass the chiral symmetry is broken. In both cases the mass of the vector state remainsfinite through the transition to the parity broken phase, where as mpi and mpi2 approach zero at thetransition.4. ConclusionsWe presented a lattice study of an NJLmodel, via Wilson fermions. These studies are relevantbecause four fermion interactions occur naturally in several extensions of the Standard Model.They represent low energy parameterisations of a more fundamental theory. Additionally modelsof dynamical electroweak symmetry breaking make extensive use of four-fermion operators toendow the Standard Model fermions with masses. We investigated spontaneous chiral symmetrybreaking for the lattice version of the NJL model presented here.One observes that disconnected contributions to the correlator of the pseudoscalar meson be-come significant at large coupling. Evaluating the disconnected contribution requires significantlymore computational effort than evaluating the connected part. At relatively small mass and largecoupling the correlator of the auxiliary field pi can be used instead.6Four Fermion Interactions with Wilson Fermions Jarno RantaharjuAccording to the expectations we have find evidence of spontaneous chiral symmetry breakingat large coupling, γ = 0.65a and 0.6a. The results are compatible with previous studies and followsmeanfield results [7–9, 14]. Additional work is needed to cover the full phase space of the modeland to verify scaling and lattice size dependence. Once these analyses are performed it is thennatural to move to the full gauged model.5. AcknowledgmentsWe thank Michele Della Morte for useful discussion. This work was supported by the DanishNational Research Foundation DNRF:90 grant and by a Lundbeck Foundation Fellowship grant.The computing facilities were provided by the Danish Centre for Scientific Computing.References[1] J. Krog, M. Mojaza and F. Sannino, Phys. Rev. D 92, 085043 (2015) [arXiv:1506.02642 [hep-ph]].[2] W. A. Bardeen, C. T. Hill and M. Lindner, Phys. Rev. D 41, 1647 (1990).[3] G. Cacciapaglia and F. Sannino, arXiv:1508.00016 [hep-ph].[4] B. Holdom, Phys. Rev. D 24, 1441 (1981).[5] H. S. Fukano and F. Sannino, Phys. Rev. D 82 (2010) 035021 [arXiv:1005.3340 [hep-ph]].[6] K. Yamawaki, hep-ph/9603293.[7] K. M. Bitar and P. M. Vranas, Phys. Lett. B 327 (1994) 101 [hep-lat/9310008].[8] K. M. Bitar and P. M. Vranas, Phys. Rev. D 50 (1994) 3406 [hep-lat/9310027].[9] S. Aoki, S. Boettcher and A. Gocksch, Phys. Lett. B 331 (1994) 157 [hep-lat/9312084].[10] A. Hasenfratz, P. Hasenfratz, K. Jansen, J. Kuti and Y. Shen, Nucl. Phys. B 365 (1991) 79.[11] A. Ali Khan, M. Gockeler, R. Horsley, P. E. L. Rakow, G. Schierholz and H. Stuben, Nucl. Phys. Proc.Suppl. 34 (1994) 655 [hep-lat/9401003].[12] J. B. Kogut, J. F. Lagae and D. K. Sinclair, Phys. Rev. D 58 (1998) 034504 [hep-lat/9801019].[13] D. K. Sinclair, J. B. Kogut and J. F. Lagae, Nucl. Phys. Proc. Suppl. 73 (1999) 471 [hep-lat/9809052].[14] S. Catterall, R. Galvez, J. Hubisz, D. Mehta and A. Veernala, Phys. Rev. D 86 (2012) 034502[arXiv:1112.1855 [hep-lat]].[15] M. Bochicchio, L. Maiani, G. Martinelli, G. C. Rossi and M. Testa, Nucl. Phys. B 262 (1985) 331.[16] T. Izubuchi, J. Noaki and A. Ukawa, Phys. Rev. D 58 (1998) 114507 [hep-lat/9805019].7

Wilson Fermions with Four Fermion Interactions

Abstract

Similar works

Full text

Available Versions

Plymouth Electronic Archive and Research Library