Baryon Physics in Holographic QCD Models (and Beyond) in collaboration with A. Wulzer

Transcription

1 Baryon Physics in Holographic QCD Models (and Beyond) Alex Pomarol (Univ. Autonoma Barcelona) in collaboration with A. Wulzer

2 One of the main goals of AdS/CFT: Find the string dual theory of QCD One of the closest example: Sakai-Sugimoto Model (D4/D8 system) Dual realization of Chiral symmetry breaking: z SU(N F ) L SU(N F ) R SU(N F ) V

3 At large N and large thooft coupling, g²n in the 4D gauge theory, the string dual theory is, at low-energies, a weakly-coupled gauge theory in 5D with chiral symmetry breaking on the z=0 boundary UV-bound. AdS 5 IR-bound. SU(2) L U(1) Y SU(NSO(5) F ) L SU(N U(1) F ) R SU(N F ) V SO(4) U(1) Fermions 5 of SO(5) z = z =0 warped extra dim: z Dirichlet: boundy conditions: A µ L Aµ R =0 Neumann: F µ5 L + F µ5 R =0

4 Proposed Holographic QCD model of two massless quarks Erlich,Katz,Son,Stephanov Da Rold,A.P.; Hirn,Sanz UV-bound. AdS 5 IR-bound. SU(2) L U(1) Y SU(N F SO(5) ) L SU(N U(1) F ) R U(2) L U(2) R SU(N F ) V U(2) V SO(4) U(1) z =0 Fermions 5 of SO(5) warped extra dim: z AdS space z = L 1 Dirichlet: boundy conditions: A µ L Aµ R =0 Neumann: F µ5 L + F µ5 R =0

5 5D Lagrangian: ds 2 = a(z) 2 [dx 2 + dz 2 ] a(z) M N c 64π 2 ] [Tr[L MN L MN ]+ α2 2 ˆL ˆLMN MN + {L R} ɛ MNOP Q LM Tr[L NO L PQ ] {L R} +... Chern-Simons term: Needed to reproduce the U(1)-anomaly in QCD Coefficient fixed no extra parameter!

6 5D lagrangian: a(z) M N c 64π 2 ] [Tr[L MN L MN ]+ α2 2 ˆL ˆLMN MN + {L R} ɛ MNOP Q LM Tr[L NO L PQ ] {L R} +... Theory of 3 parameters: M 5,L 1,α Compactification scale: Mass gap Model that successfully describes many properties of QCD mesons (Kaluza-Klein states)

7 Baryon sector?

8 Baryon sector? In the large- N c limit: M B N c = 1 1/N c Baryon are solitons! Witten Original motivation for Baryon as Skyrmions: solitons of the chiral lagrangian Skyrme 61, Adkins+Nappi+Witten 83

9 If only the F²-term is considered... Baryons: Solitons of the 5D theory instanton in 4D ( t E z ) Topological charge: Q = 1 16π 2 d 3 x L1 0 [ dz T r Lˆµˆν Lˆµˆν Rˆµˆν Rˆµˆν ] = N inst (L) N inst (R) ˆµ =1, 2, 3, 4(L 0 = R 0 = 0) Atiyah,Manton89 Son,Stephanov04 reduces to 4D Skyrmion charge Q = 1 24π 2 d 3 x ɛ ijk Tr [ U i U U j U U k U ] Z. U(x): pion field

10 Is the Baryon given by the 4D instanton configuraton? Yes, in an infinite flat space. Energy independent of size: E E =8π 2 M 5 ρ size

11 Is the Baryon given by the 4D instanton configuraton? Yes, in an infinite flat space. Energy independent of size: E E =8π 2 M 5 ρ size But curvature and compactification breaks the scale invariance of the instanton: E [ E =8π 2 M 5 1+ ρ ] 2R ρ size

12 Is the Baryon given by the 4D instanton configuraton? Yes, in an infinite flat space. Energy independent of size: E E =8π 2 M 5 ρ size But curvature and compactification breaks the scale invariance of the instanton: E [ E =8π 2 M 5 1+ ρ ] 2R ρ size It shrinks to zero size! Its stability is UV-sensitive: Depends on the higher-dimensional operators of the theory

13 Lowest higher-dimensional operator: Dimension five-operator: Chern-Simons term

14 Lowest higher-dimensional operator: Dimension five-operator: Chern-Simons term Stability and consistency of the model: 5D model: L 5 F Λ 5 AF F + 1 Λ 4 5 F Soliton energy E(ρ) ρm KK + 1 ρ 2 Λ 2 5 ρ 1 M 1/3 KK Λ2/3 5 >> 1 Λ 5 No sensitive to higher-dim operators Different from the 4D Skyrmion model: 4D Skyrmion: L χ (D µ U) m 2 ρ (D µ U) ρ 1 m ρ Sensitive to higher-dim operators

15 Mainly two approaches towards Holographic Baryons: a) Treat baryons as instanton configurations H.Hata,T.Sakai,S.Sugimoto,S.Yamato; D.K.Hong,T.Inami,H.U.Yee; D.K.Hong, M.Rho,H.U. Yee,P.YH.Hata,M.Murata,S.Yamato; K.Hashimoto,T.Sakai,S.Sugimoto b) Find the new 5D soliton configuration including the curvature of the space and the CS-term A.P., Wulzer; Wulzer, Panico

16 Mainly two approaches towards Holographic Baryons: a) Treat baryons as instanton configurations H.Hata,T.Sakai,S.Sugimoto,S.Yamato; D.K.Hong,T.Inami,H.U.Yee; D.K.Hong, M.Rho,H.U. Yee,P.YH.Hata,M.Murata,S.Yamato; K.Hashimoto,T.Sakai,S.Sugimoto b) Find the new 5D soliton configuration including the curvature of the space and the CS-term A.P., Wulzer; Wulzer, Panico Several problems: Not fully consistent (CS-term is sizable) Does not reproduce large-n expectations Wulzer,Panico; A.Cherman,T.D. Cohen,M.Nielsen

17 Solution of the SU(2) part: Ansatz (Witten 77): cylindrical symmetry invariance under the combine SU(2) gauge and rotation + Parity: L a j = 1+φL 2 (r, z) r 2 ɛ jak x k + φl 1 (r, z) ( r 2 ) A L r 3 δ ja x j x a + 1 (r, z) r 2 x j x a L a 5 = AL 2 (r, z) x a r R a j (x, z) = L a j ( x, z) R a 5(x, z) =L a 5( x, z)

18 Solution of the SU(2) part: Ansatz (Witten 77): cylindrical symmetry invariance under the combine SU(2) gauge and rotation + Parity: L a j = 1+φL 2 (r, z) r 2 ɛ jak x k + φl 1 (r, z) ( r 2 ) A L r 3 δ ja x j x a + 1 (r, z) r 2 x j x a L a 5 = AL 2 (r, z) x a r R a j (x, z) = L a j ( x, z) R a 5(x, z) =L a 5( x, z) E = 16π 0 dr zir z uv dz M 5 a(z) 4 fields: combine in a 2D gauge boson + a complex scalar 2D Abelian Higgs models: [ 1 2 D µφ r2 F 2 µ ν + 1 ( ) ] 1 φ 2 2 4r 2.

19 Solution of the U(1) part: Ansatz: ˆL 0 = ˆR 0 = 1 α s(r, z) r

20 Solution of the U(1) part: Ansatz: ˆL 0 = ˆR 0 = 1 α E =8πM 5 0 dr s(r, z) r L1 0 dz scalar coupled to the topological charge density of the 2D Abelian model [ a(z) 1 2 ( µs) 2 πγ s ] r ρ topo 1/r 2 Creates a potential that prevents the shrinking of the soliton

21 2D Abelian Higgs model + Scalar Total: 5 fields System of 5 non-linear PDEs in 2D + suitable b.c. to ensure Q=1 Solution must be found numerically We rely on FEMLAB (COMSOL) package used by engineers in many physical systems

22 Skyrmion energy density E = drdz ρ E = f(m 5,L 1,α) 1140 MeV z r taking F π,m ρ,f ω /F ρ to fix the 3 parameters a instanton-like skyrmion, without the CS, will have 1/2 of the energy

23 Identification of the the proton and neutron: Standard Procedure: Adkins+Nappi+Witten 83 1) Identify the time-dependent fluctuations of the rotational zero-modes: SU(2)-rotation of the soliton doesn t change its energy U = a 0 + i σ a i SU(2) a 0,a i : collective coordinates (quantum mechanical variables) 2) Calculate H 3) Calculate the spin and isospin operator 4) Eigenstates of spin and isospin 1/2 [Pl') = 1 (al + ia2), [p$) = - / ( a ia3), In?) = / (ao + ia3), In,l,) = - % ( a l -- ia2),,/7- qt / _

24 Extra difficulty: After turning on a 0,i (t) we must assure that the EOM are satisfied: Fields that were zero in our Ansatz turn on

25 Static properties of the baryon Baryon couplings to external sources: B J µ B = J boundary µ soliton J boundary µ = δl 5D δa non norma µ = F 5µ boundary Axial coupling, magnetic and electric form factors (and moments) can be calculated Tree-level: G S E = N c dr r j 0 (qr)(a(z) z s) 6πγL UV G V E = 4πM [ )] 5 dr r 2 j 0 (qr) a(z) ( z v 2(D z χ) 3λ (2) UV G S M = 8πM NM 5 α dr r 3 j 1(qr) (a(z) z Q) 3λ qr UV G V M = M N N c dr r 2 j ) 1(qr) (a(z)(d z φ) 3πLγα qr (2) UV G A = M [ N N c dr r a(z) j ) 1(qr) ((D z φ) E 3παγL qr (1) ra zr ] a(z)(d z φ) (1) j 0 (qr) ( UV

26 Results: A.P.,Wulzer; Wulzer,Panico Experiment AdS 5 Deviation M N 940 MeV 1130 MeV 20% µ S % µ V % g A % re,s fm 0.88 fm 11% re,v fm rm,s fm 0.92 fm 12% rm,v fm r 2 A 0.68 fm 0.76 fm 12% µ p /µ n %

27 Results: A.P.,Wulzer; Wulzer,Panico Experiment AdS 5 Deviation M N 940 MeV 1130 MeV 20% µ S % µ V % g A % re,s fm 0.88 fm 11% re,v fm rm,s fm 0.92 fm 12% rm,v fm r 2 A 0.68 fm 0.76 fm 12% µ p /µ n % Large deviations: Possible due to have a massless pion Due to have a massless pion Very small deviation: Prediction including subleading large-n corrections µ p /µ n = (µ V + µ S )/(µ V µ S ) 1 2µ S /µ V.

28 q 2 GeV 2 Form factors: Dashed line: Empirical dipole fit Figure 1: Scalar (left) and vector (right) electric the empirical dipole fit (dashed Wulzer,Panico line) [7] G E S q G M S q V q 2 GeV 2 q 2 GeV 2 1 G M P q 2 GM N q q 2 GeV Figure 2: Normalized scalar (left) and vector (rig 1.0 results with the empirical dipole fit (dashed line does not invalidate the general picture. G A q 2 G A It is interesting to notice that a much bette 0.4 instead of the standard procedure [6] considered 0.2 quantization of collective coordinates of the skyr The results of Ref. [17] can be directly applied q 2 GeV 2 collective coordinate quantization, the 5D natur

29 Goldberger-Treiman relation In large N: A µ π N N g A = F πg πnn M N Also fulfilled in this model

30 Baryons in Warped 5D models for Electroweak Symmetry breaking a) Higgsless: 5D version of Technicolor F π 246 GeV M KK 1.2 T ev b) Higgs as Pseudo-Goldstone boson (PGB) ~ pion F π 500 GeV M KK 2.4 T ev If stabilized by a CS-term as in 5D QCD case, Baryon mass: M B 6 T ev ( F 246 GeV ) 2 ( 1 T ev M KK ) Out of the reach at the LHC! Higgsless Higgs as PGB 5 TeV 10 TeV

31 But we have more freedom in EWSB models: Baryon ~ Instanton configuration stabilized by a small effect coming from a higher-dim operator Mass: M B 8π 2 M 5 (model-independent) They could be lighter (2-3 TeV) and smaller (ρ~1/tev)

32 Cosmological Implications: Stable states (they carry topological charges) Potentially Dark Matter candidates Small energy densities if thermally produced: (They must be neutral under the SM group) For σ ρ 2 Ωh ( 1 T ev 1 ρ ) 2 << 0.1

33 Alternatives ways to produce them: As real baryons, we can use Baryogenesis mechanism in particular electroweak baryogenesis Production of 5D baryon at the confiniment/deconfinement phase transition Possible to study in extra dimensional models: Hawking-Page transition Heating the extra dim, metric changes to that of an AdS Black-hole (First order phase transition) Barr,Chivukula,Farhi 90 Possibility to produce these new baryon at the same time that real baryons Ω DM related with Ω B

34 Direct detection: Even if neutral under SM gauge group, its large magnetic moments allow to detect them... work in progress

35 Conclusions Baryons of a 5D QCD-like model arise as solitonic configurations We have showed that they are stable and consistently described within the 5D model We have found the exact solution (numerically) and calculated their properties (masses, couplings,...) behave like real baryons Interesting cosmological implications for models of EWSB: worthy to fully explore More to do: incorporate quark masses, study phase transitions at high baryon number,...

36

37 Anomalous couplings Meson physics from the CS term: π π N c 64π 2 ɛmnop Q γ γ γ ω d 5 x L M Tr[L NO L PQ ] {L R} N c 1 πf 48π 2 µν (γ) F π F (γ) ρσ F (γ) ρσ N c g ρππ πf 16π 2 µν (ω) F π an predict plenty of an ω :1 (I = 0) π ρ ω N c gρππ 2 x πf 16π 2 µν (ω) F π F (ρ) ρσ x 1.16 ρ :1 (I = 1)

38 Relevant for meson partial decay widths γ ω, ρ Γ(ω, ρ πγ) π ω ρ π π π Γ(ω 3π) ω ρ γ π µ(e) Γ(ω πµµ(ee)) µ(e) also anomalous form factors Grigoryan,Radyushkin

39 Experiment AdS 5 Γ(ω πγ) Γ(ω 3π) Γ(ρ πγ) Γ(ω πµµ) Γ(ω πee) Average error of ~ 10%