Lattice check of dynamical fermion mass generation

Transcription

1 Lattice check of dynamical fermion mass generation Petros Dimopoulos Centro Fermi & University of Rome Tor Vergata Workshop on the Standard Model and Beyond, Corfu, September 2-10, 2017

2 in collaboration with S. Capitani Universität Frankfurt am Main G.M. de Divitiis Università di Roma Tor Vergata R. Frezzotti Università di Roma Tor Vergata M. Garofalo Higgs Centre for Theoretical Physics, The University of Edinburgh B. Knippschild HISKP (Theory), Universität Bonn B. Kostrzewa HISKP (Theory), Universität Bonn F. Pittler HISKP (Theory), Universität Bonn G.C. Rossi Università di Roma Tor Vergata & Centro Fermi C. Urbach HISKP (Theory), Universität Bonn

3 Theoretical proposal - R. Frezzotti and G.C. Rossi Nonperturbative mechanism for elementary particle mass generation PRD 92 (2015) R. Frezzotti and G.C. Rossi Dynamical mass generation PoS LATTICE2013 (2014) 354 & other relevant publications - R. Frezzotti. M. Garofalo and G.C. Rossi Nonsupersymmetric model with unification of electroweak and strong interactions PRD 93 (2016) S. Capitani et al. Check of a new non-perturbative mechanism for elementary fermion mass generation PoS LATTICE2016 (2016) S. Capitani et al. Testing a non-perturbative mechanism for elementary fermion mass generation: lattice setup PoS LATTICE S. Capitani et al. Testing a non-perturbative mechanism for elementary fermion mass generation: Numerical results PoS LATTICE F. Pittler Spectral statistics of the Dirac operator near a chiral symmetry restoration in a toy model PoS LATTICE2017

4 Overview Widely accepted incompletness of the SM -for a number of fundamental phenomena not satisfactorily or not at all described by/within it- has motivated a vast variety of ingenous and original New Physics proposals BSM. However, this task is proved to be non-trivial perhaps due to the fact that SM is a renormalisable theory. SM describes elemenentary particle masses employing the symmetry breaking SU(2) L U(1) Y U(1) em. The hierarchy pattern of fermion masses (but also Higgs mass un-natural feature) lack deep understanding, they are rather accomodated by fitting to experimental data.

5 Overview Dynamical generation of fermion masses Similar physics effect which generates qq 0 where dynamical χsb triggered by an explicit χsb term i.e. fermion mass or Wilson term. In massless LQCD with Wilson term Non-Perturbative contribution ( Λ QCD ) is accompanied by an 1/a divergent term. Separation of the two effects requires an infinite fine tuning ( naturalness problem). Proposal: QCD extended to a theory with enriched symmetry for tackling naturalness problem.

6 Overview Dynamical generation of fermion masses owing to a NP mechanism triggered by a Wilson-like (naively irrelevant) chiral breaking term. Simplest toy-model where the mechanism can be realised: SU(N f = 2) doublet of strongly (SU(3) c ) interacting fermions coupled to scalars via Yukawa and Wilson-like terms physics depends crucially on the phase (Wigner or NG) enhanced symmetry (naturalness à la t Hooft) leads to Φ -independence of fermion masses The intrinsic NP character of the mechanism requires lattice numerical investigation of the toy model. The proposed mechanism can be falsified/verified.

7 Theoretical setup Toy-model: QCD Nf =2 + Scalar field + Yukawa + Wilson L toy = L kin (Q, A, Φ) + V (Φ) + L Y (Q, Φ) + L W (Q, A, Φ), with: L kin (Q, A, Φ) = 1 4 F µνf a µν a + Q L γ µd µq R + Q R γ µd µq L + 1 ] [ Φ 2 Tr Φ ] + 1 ( ]) 2 4 λ [ Φ Φ V (Φ) = 1 2 µ2 Tr Tr L Y (Q, Φ) = η ( QL ΦQ R + Q ) R ΦQ L L W (Q, A, Φ) = ρ b2 2 [ Φ Φ ( QL D µφd µq R + Q R D µφ D µq L ) (where D µ = µ + ig sλ a A a µ, D µ = µ ig sλ a A a µ) Q: fermion SU(2) doublet coupled to SU(3) gauge field and to scalar field through Yukawa and Wilson terms. b 1 : UV cutoff.

8 Theoretical setup (contd.) χ L χ R transformations are symmetry of L toy : χ L : χ L (Φ Ω L Φ) χ R : χ R (Φ Ω R Φ) χ L : Q L Ω L Q L, χ R : Q R Ω R Q R, Q L Q L Ω L Ω L SU(2) L Q R Q R Ω R Ω R SU(2) R Exact symmetry χ χ L χ R acting on fermions and scalars NO power divergent mass terms. The (fermion) χ χ L χ R transformations are not a symmetry for generic (non-zero) η and ρ. P, C, T, gauge invariance... are symmetries & power counting renormalisation.

9 Theoretical setup (contd.) The shape of V (Φ) determines crucially the physical implications of the model When the scalar potential V (Φ) has one minimum χ L χ R is realized à la Wigner. The (fermion) χ χ L χ R transformations generate Schwinger-Dyson Eqs (unrenormalised). They get renormalised after considering the operator mixing procedure. PT operator mixings NO χ SSB phenomenon occurs NO NP fermion mass generation

10 Theoretical setup (contd.) Critical Model: χ-symmetry restoration occurs when the Yukawa term is compensated by the Wilson term. This takes place (in the Wigner phase) at a certain value of the Yukawa coupling. L,i R,i In fact, for J µ (or J µ ) get µ Z J JL,i µ (x)o(0) = (η η(η; g0 2, ρ, λ)) [ Q L τ i ΦQ R h.c.](x)o(0) + O(b 2 ) (SDE renrm/tion here analogous to chiral SDE renrm/tion in Bochicchio et al. NPB 1985) L,i R,i enforce the current J µ (or J µ ) conservation = η η(η; g0 2, ρ, λ) = 0 ηcr (g 0 2, ρ, λ). The Low-Energy effective action (in the Wigner phase) reads Γ Wig µ 2 Φ >0 = 1 4 (F F ) + Q /DQ + (η η cr )( Q L ΦQ R + h.c.) Tr [ µφ µφ ] + V µ 2 Φ >0 (Φ) in the critical theory ( χ is a symmetry, up to O(b 2 )) Scalars decoupled (up to cutoff effects) from quarks and gluons. no fermionic mass (m Q = 0 up to O(b 2 )).

11 Numerical investigation Lattice simulation details Lattice discretization, L latt., with exact χ-symmetry. Use naive fermions with symmetric covariant derivative, µ, throughout. We limit our first study to the quenched approximation Quenching: independent generation of gauge (U) and scalar (Φ) configurations. it is quite certain that the mechanism under investigation, if confirmed, survives quenching. Naive fermions are relatively cheap and fine with quenched approximation. (For an unquenched study one might employ staggered or domain-wall or overlap fermions)

12 Numerical investigation Lattice simulation details To avoid exceptional configurations ( due to fermions zero modes) introduce twisted mass regulator L latt. +iµ Q Qγ5τ 3 Q. (Frezzotti, Grassi, Sint and Weisz, JHEP 2001) at a cost of soft breaking of χ L χ R, symmetry recovered after an extrapolation to µ Q 0. Locally smeared Φ in QD lat [U, Φ]Q for noise reduction.

13 Numerical investigation Lattice simulation parameters simulations at two values of the lattice spacing β = 5.75 (b = 0.15 fm) & β = 5.85 (b = 0.12 fm ) L/b = 16 & T /b = 40 use lattice scale r 0 = 0.5 fm (motivated from QCD, for illustration) Guagnelli, Sommer and Wittig NPB 535 (1998) & Necco and Sommer NPB 622 (2002) ρ = 1.96 in the Wigner & NG phase (for checking the validity of the mechanism it is sufficient to set some reasonable value 0) choose scalar field parameters by imposing conditions on (r 0M σ) 2, λ R and (r 0v R ) 2 statistics: #configs (gauge scalar) 240 several values of the Yukawa coupling η (and µ Q ).

14 Determination of η cr in the Wigner phase Compute correlation function where V 3 C J D (x y) J 0 (x) D S3 (y) J 0 V 3 (x) = J 0 L 3 (x) + J 0 R 3 (x) [ τ 3 QL/R (x ˆ0)γ 0 2 U0(x ˆ0)Q L/R (x) + Q τ 3 L/R (x)γ 0 D S3 (y) = Q L (y) [Φ, τ ] [ ] 3 Q R (y) 2 Q τ 3 R (y) 2, Φ Q L (y) J L/R 3 0 (x) = 1 2 ] 2 U 0 (x ˆ0)Q L/R (x ˆ0) Renormalised Schwinger-Dyson eqs of Ṽ 3 -type (in the form of a would be χ-wti): µ J V 3 µ = (η η cr ) D S3 + O(b 2 ) and 0 0 J V 3 0 M S f MS M 2 M S

15 Determination of η cr in the Wigner phase In the Wigner phase at η = η cr the restored χ-symmetry is realised à la Wigner (first example in a local setting) and leads to vanishing correlation function V 3 C J D (x y) J 0 (x) D S3 (y) i.e. vanishing matrix element: ME = 0 J V 3 0 M S M S D S3 0 η ηcr 0 J 0 V3 (x) D S3 (y) 1.5e e e e e e-04 η = η = η = η = η = e t/b (example at β = 5.85 and aµ Q = )

16 Determination of η cr in the Wigner phase An accurate determination of η cr is obtained employing the WI ratio i.e. compute: r WI = 0 J V 3 0 (x) D S3 (y) D S3 (x) D S3 (y) at several values of η (and µ Q ) and extrapolate to r WI 0 :

17 Determination of η cr in the Wigner phase An accurate determination of η cr is obtained employing the WI ratio i.e. compute: r WI = 0 J V 3 0 (x) D S3 (y) D S3 (x) D S3 (y) at several values of η (and µ Q ) and extrapolate to r WI 0 : 0.0e+00 [β = 5.75] ηcr = 1.272(5) 0.0e+00 [β = 5.85] ηcr = 1.208(4) rwi -2.0e-02 rwi -2.0e e e e η e η -1.0 ( r WI in latt. units after extrapolating to aµ Q = 0)

18 Determination of η cr in the Wigner phase An accurate determination of η cr is obtained employing the WI ratio i.e. compute: r WI = 0 J V 3 0 (x) D S3 (y) D S3 (x) D S3 (y) at various values of η (and µ Q ) and extrapolate to r WI 0 : 0.0e+00 [β = 5.75] ηcr = 1.272(5) 0.0e+00 [β = 5.85] ηcr = 1.208(4) rwi -2.0e-02 rwi -2.0e e e e η e η -1.0 η cr = β = 5.75 & η cr = β = 5.85 A few per mille statistical error for η cr determination. (Comparable systematic uncertainties - Preliminary results!)

19 Features and properties of the toy-model in NG-phase V (Φ) of mexican hat shape χ L χ R realised à la NG. χ L χ R spontaneously broken: Φ = v + σ + i τ π, Φ = v 0. ( ) L W (Q, A, Φ) = ρb2 r bvρ ( 2 Q L D µφd µq R + h.c. LQCD ar W (Q, A) = 2 Q L D 2 Q R + h.c. ). In the critical theory η = η cr : a b the (Yukawa) mass term, v QQ, gets cancelled. χ breaking due to residual O(b 2 v) effects is expected to trigger dynamical χsb.

20 Features and properties of the toy-model in NG-phase Look for dynamically generated fermion mass: NP mass term has to be χ L χ R invariant (and under chiral variation can be accomodated in the χ WTI s). Note that a term like m[ Q L Q R + Q R Q L ] is not χ L χ R invariant. At generic η, two χ breaking operators are expected to arise: Yukawa induced + dynamically generated ( conjecture) Γ NG =... + (η η cr )( Q L Φ Q R + h.c.) + c 1Λ s( Q L UQ R + h.c.) where U = Φ Φ Φ = (v + σ) 1 + i τ ϕ τ ϕ 1 + i v 2 + 2vσ + σ 2 + ϕ ϕ v +... and Λ s RGI NP mass scale. U is a non-analytic function of Φ, but transforms like Φ under χ L χ R ; obviously U can not be defined in the Wigner phase ( Φ = 0) no NP mass or mixings in the Wigner phase. Note that (χ-inv. term): c 1Λ s( Q L UQ R + h.c.) c 1Λ s QQ +...

21 Features and properties of the toy-model in NG-phase Work at the same lattice parameters (β, λ, ρ and volume) as in the Wigner phase A± Compute WTI quark mass: m WTI 0 J 0 (x)p ± (y) = in the NG-phase (where P ± (x)p ± (y) P ± = Qγ 5τ ± Q - pseudoscalar density) & M ps from P ± (x)p ± (y).

22 Features and properties of the toy-model in NG-phase Work at the same lattice parameters (β, λ, ρ and volume) as in the Wigner phase M PS 0 At η = η cr : m WTI 0, Note m WTI η = η cr = (η η cr )v + c 1Λ s m WTI = c 1Λ s

23 Features and properties of the toy-model in NG-phase Work at the same lattice parameters (β, λ, ρ and volume) as in the Wigner phase M PS 0 At η = η cr : m WTI 0, Note m WTI η = η cr = (η η cr )v + c 1Λ s m WTI = c 1Λ s m WTI cancels at η = η cr c 1Λ s/v η cr η c 1Λ s 0 At η = η cr for β = 5.85: M PS 320 MeV and m WTI bare 0. (Preliminary results!)

24 Features and properties of the toy-model in NG-phase Work at the same lattice parameters (β, λ, ρ and volume) as in the Wigner phase M PS 0 At η = η cr : m WTI 0, Note m WTI η = η cr = (η η cr )v + c 1Λ s m WTI = c 1Λ s m WTI cancels at η = η cr c 1Λ s/v η cr η c 1Λ s 0 At η = η cr for β = 5.75: M PS 325 MeV and m WTI bare 0. (Preliminary results!)

25 Towards the CL: scaling behaviour (r 0M PS) 2 against η at two β-values:

26 Towards the CL: scaling behaviour (r 0M PS) 2 against η at two β-values: M PS : (very) small cutoff effects M PS 0 at CL At η = η cr : m WTI : renormalised may differ by 5-10% wrt bare, but observed small cutoff effects quite certain m WTI 0 at CL

27 Towards the CL: scaling behaviour c 1Λ S 0 (η η cr ) 0 (η η cr ) has to be renormalised... Consider renormalised quantity: D η = (η η cr )d(r 0M PS ) 2 /dη ηcr Dη (b/r 0 ) 2 Quite certainly D η 0 at CL

28 Conclusions & Outlook We have presented a toy-model that exemplifies a novel NP mechanism for fermion mass generation. R. Frezzotti and G.C. Rossi, PRD 2015, [arxiv: [hep-lat]] The toy model is a non-abelian gauge model with an SU(N f = 2)-doublet of strongly interacting fermions coupled to scalars through Yukawa and Wilson-like terms: at the critical point, where (fermion) χ invariance is recovered in Wigner phase (up to UV-effects) the model is conjectured to give rise in NG phase to dynamical χ-ssb and hence to non-perturbative fermion mass generation. The main physical implications of the conjecture above can be verified/falsified by numerical simulations of the toy-model (rather cheap in the quenched approximation).

29 Conclusions & Outlook A study at two values of the lattice spacing ( 0.12 and 0.15 fm) in the quenched approximation has been presented. We have shown that the critical value of the Yukawa coupling in the Wigner phase at which χ is restored can be accurately determined. Then we explored the effects of dynamical SSB of the (restored) χ-symmetry in the NG phase which look very well compatible with the generation of a non-zero (effective) fermion mass and M PS O(Λ s ) at the CL. These findings might be checked and verified at a finer value of the lattice spacing in order to get even more solid confirmation for the persistence of the dynamical mass generation mechanism in the continuum limit.

30 Conclusions & Outlook A study at two values of the lattice spacing ( 0.12 and 0.15 fm) in the quenched approximation has been presented. We have shown that the critical value of the Yukawa coupling in the Wigner phase at which χ is restored can be accurately determined. Then we explored the effects of dynamical SSB of the (restored) χ-symmetry in the NG phase which look very well compatible with the generation of a non-zero (effective) fermion mass and M PS O(Λ s ) at the CL. These findings might be checked and verified at a finer value of the lattice spacing in order to get even more solid confirmation for the persistence of the dynamical mass generation mechanism in the continuum limit. Thank you for your attention!

31 Extra slides

32 Eucledian time behaviour for M eff (t) associated to the correlation function C J D(x y) J V 3 0 (x) D S3 (y) & r WI (example case η = aµ Q = in Wigner phase & β = 5.85): e e-01 Meff(t/b) brwi -1.4e e e t/b e t/b

33 Extrapolation in µ Q = 0 and in η of r WI = 0 J V 3 0 (x) D S3 (y) D S3 (x) D S3 (y) in order to determine η cr where r WI (in latt. units) vanishes (β = 5.85): 0.0e e-02 η = η = η = e+00 ηcr = 1.208(4) rwi -2.0e e-02 rwi -2.0e e e e e e-04 (aµq) 2 1.0e e e η -1.0

34 Extrapolation in µ Q = 0 (left panel) and η-dependence (right panel) for 2r 0 m WTI = 2r J A± 0 M PS ± 0 P ± in the NG-phase (β = 5.85). M PS ± (2r0m WTI ) η = η = η = η = (r0µq)

35 Extrapolation in µ Q = 0 (left panel) and η-dependence (right panel) for (r 0 M PS ) 2 (β = 5.85) (r0mps) η = η = η = (r0µq)