Hadronic Superpartners from Superconformal and Supersymmetric Algebra

Transcription

1 Hdronic Superprtners from Superconforml nd Supersymmetric Algebr mesons tetrqurks bryons Mrin Nielsen SLAC & IFUSP with S.J. Brodsky, rxiv:

2 Light Front Hologrphic QCD (LFHQCD) Brodsky, de Térmond, Dosch (rxiv: ) certin spects of quntum field theory in four spce-time dimensions cn be obtined s limiting vlues of five dimensionl theory front form Light Front instnt form AdS/QCD This effective theory follows from the clustering properties of the LF Hmiltonin nd it's hologrphic embedding in AdS 5 spce. The resulting theory leds to supersymmetric one-dimensionl hdronic LF boundstte equtions for mesons, bryons nd tetrqurks, providing semiclssicl pproximtion to strongly coupled QCD dynmics

3 AdS 5 Action for free sclr (pseudo-sclr) field tensor metric in AdS 5 with spce curvture defining A(z) = log z + log R ) g MN = e 2A(z) MN ; p g = e 5A(z) Euler-Lgrnge eqution, =1,..4; pple =3 interction term Fourier trnsforming nd scling

4 Schrodinger-like eqution S(q, z) =z 3/2 (q, z), B=0 for regulr solutions t z=0 Boundry conditions { hrd wll: Lgrngin is defined only for z<z 0 J ν (qz 0 )=0 soft wll: Lgrngin is modified by dilton term e φ(z) zeros of the Bessel functions led to M 2 =q 2 not comptible with experimentl dt M= q 2, '(z) = z 2 new term introduces scle!

5 , { 2D hrmonic oscilltor Hmiltonin with ngulr momentum L M 2 nl = q 2 =2E nl (pple 1) =4 n + L AdS 2, for pple =3, > 0 the lowest pseudo-sclr prticle (n=0, L=0) hs M nl =0 s expected in the chirl limit

6 AdS 5 Lgrngin for free vector field sme s for the sclr field but k=1 Euler-Lgrnge eqution fter Fourier trnsforming nd scling sme solutions but with L 2 AdS =1+(µR) 2 Mesons with rbitrry spin J 2 z sme s before but

7 rescling:, leds to the Schrodinger-like eq., q 2 = M 2 nl AdS J =4 n + L AdS + J 2, > 0 Light Front Hmiltonin x P mss of 2-constituent prticle in LF (1-x)P for m i =0, x nd b cn be expressed in term of the LF vrible:

8 construct the LF Hmiltonin:, defining:, where L is the LF ngulr momentum, we get fter rescling LF potencil Light Front Hologrphic QCD (LFHQCD) compring with the AdS 5 Schrodinger eq.:, q 2 = M 2 nlj =4 Regge trjectories only 1 prmeter n + L + J 2, for > 0 the pion (n=l=j=0) hs 0 mss!

9 for J = L + S ) M 2 n,l,s =4 n + L + S 2 poor greement! (S=0) smll qurk mss correction ws included! (S=1)

10 AdS 5 Action for hlf integer spin prticles described by spinor with tensor indices, T=J-1/2 Yukw like term insted of the dilton term since dilton term cn be bsorbed in the fermion field nd does not led to interction Euler-Lgrnge eqution Fourier trnsforming, scling nd in terms of the chirl components Ψ +, Ψ -

11 compring with the meson eq.: leds to: M 2 = M 2 n,l =4 B (n + L + 1), with The two chirl components of the fermion hve different ngulr momentum

12 Light Front Effective Fermionic Hmiltonin The light-front wve eqution describing bryons is mtrix eigenvlue eqution d D LF = i d + +1/2 D LF i = M i, with H LF = DLF 2 nd + V ( ) Dirc mtrices 2X2 chirl spinor : i = + )

13 LFHQCD compring with AdS 5 eqs.: z $ L AdS $ U ± $ 2 B 2 +2 B ( ± 1 2 ) M 2 = M 2 n =4 B (n + + 1) V ( ) = B + ( ) $ L n( B 2 ), ( ) $ L +1 n ( B 2 ) Z d + ( ) 2 = Z d ( ) 2 = 1 2 bryon with L: the L of the chirlity + component

14 M B 2 =4 B (n + + 1) spectrum does not depend on J! N! L =0, J =1/2! L =0, J =3/2 b = L L = L + 1 2

15 M M 2 =4 M mesons (J=L+S) n + L M + S 2 bryons M 2 B M=4 2 B =4 B n + B (n L+ B +1+ S + 1) 2 = L + S 2 LM=LB +1, S diqurk spin N : L =0, S =0) J =1/2 : L =0, S =1) J =3/2 p =0.53 GeV p =0.53 GeV S=0 S=1

16 strnge sector including smll qurk mss correction S=0 S=1 Why? S=0 S=1

17 Supersymmetric Light Front Hologrphic QCD Conforml Quntum Mechnics, DFF, Nuovo Cim. A34 (76) 569: theory invrint under trnsltions, dilttions (D) nd specil conforml (K) trnsformtions, nd obey lgebr of 1d conforml group: ny combintion of these opertors: G=uH+vD+wK, leves the ction, but not the Lgrngin, invrint! Z dt { A = L d = G [q( ), q( )] u + vt +!t defining 2 L G = 1 q 2 + q 2 g 2 4 q 2 q( ) = Q(t) p u + vt +!t 2 = v 2 4u! new term scle

18 H G (q, q) = 1 2 q 2 4 q2 + g q 2 = G(Q, Q) =uh + vd +!K Q(0)! x, Q(0)! i d dx LFHQCD Hmiltonin: ζ u=2, v=0, ω=2λ 2 (Δ<0), g=l 2-1/4, plus constnt term 2λ(J-1) justifies the form of the diltion term: it preserves conforml symmetry Wht bout meson-bryon symmetry?

19 Supersymmetric Superconforml QM (Fubini & Rbinovici, NPB245 (84) 17) grded lgebr of two fermionic opertors (super chrges) Q, Q with minimum conforml reliztion -> prticle with 2 degrees of freedom with: + f, Q + f x, spinor opertors with {, } = I,[, ] = 3 in mtrix nottion H opertes on two component sttes with sme eigenvlue: H i = E i) Symmetry!

20 effective bryon number opertor N B i = M B = 0 B Φ M, Φ B with bryon number: Supersymmetry! no scle extension to superconforml lgebr introducing the superchrges S, S, with the commuttion reltions: {S, S } = K, {Q, S } + {Q,S} =2fI 3, {Q, S } {Q,S} =4iD K nd D the sme opertors s before. In mtrix nottion:

21 introducing the new superchrge: Rλ=Q+λS G = {R,R } = {Q, Q } + 2 {S, S } + {Q, S } + {S, Q } G = H + 2 K + (2fI 3) sme form s G=uH+vD+ωK (u=1, v=0, ω=λ 2, with Δ=-λ 2 <0) plus constnt term In mtrix nottion: { G i = scle introduced in the superconforml lgebr Spectr of both opertors re identicl: Supersymmetry G G 22 M B = M 2 nf i, M 2 nf =4 n + f + 1 2

22 Supersymmetric LFHQCD Dosch, de Térmond, Brodsky, PRD91 M compring with the LFHQCD hmiltonin for meson: z 2 =M 4z 2 2 L=L M =J=f+1/2 LFHQCD hmiltonin for the bryon positive chirlity component: B L=L B =f-1/2 mesons (L M ) nd bryons (L B =L M -1) re supersymmetric prtners! L i = M MM 2 =4 (n + L M ) + only one λ L M 1 MB 2 =4 (n + L B + 1)

23 dt λ=0.53 MeV prediction π-n fmily S M =S D =0 Wht bout ρ(s M =1)- Δ(S D =1)? L L+S/2, S meson or diqurk spin const. term M 2 M =4 (n + L M ) only one λ { M 2 B =4 (n + L B + 1) G SUSY = {R,R } +2 SI M 2 M =4 M 2 B =4 n + L M + S M 2 n + L B + S D 2 +1

24 S M =1 S D =1 M 2 M =4 M 2 B =4 n + L M + S M 2 n + L B + S D 2 +1 mesons with L M =0 hve no superprtners π (L M =S M =0) M π =0 in the chirl limit

25 Tetrqurks BdTD&Lorcé, rxiv: consider the meson wve-function: z one cn show tht: ψ + n,l-1 ψ + 0 with: { N B N B + n,l i = + M n,li =0 n,l i n,l-1 meson bryon Wht bout n,lb +1? n,l B +1 = n,l B +1

26 meson bryon bryon tetrqurk S M = S D S=0 i =, G SUSY i = M 2 i tetrqurk-bryon superprtners: L T = L B originl diqurk with S D {new nti-diqurk with S = 0

27 Unrveling the qurk structure of the hdrons M 2 nlj(ads) =4 n + L + J 2 == 4 J=L+S n + L + S 2 SUSY - mesonic qq sttes J=L+S Meson (n=s M =0) Bryon Bryon (S D =0) 1 N(940) N3-/2(1520) meson-bryon superprtners L B =L M -1, S D =S M Meson (n=0,s M =1) Bryon (S D =1)

28 bryon-tetrqurk superprtners L T =L B, diqurk S=S D, ntidiqurk S=0 S D S D S=0 P(940) L B =0 [u d] u [ud]: good-diqurk S D =0, I D =0, color tetrqurk L T =L B =0 [u d] [ū d] good-diqurk, good-ntidiqurk S T =0, I T =0 prity P=(-1) L =+ (2 ntiqurks) prticle ntiprticle chrge conjugtion C=+ J PC =0 ++ cndidtes sclr sttes: J PC =0 ++, I=0

29 Jffe (PRD15(77)) sclr mesons tetrqurk sttes I=0: I=1: sme mss: esy to explin in the 4-qurk scenrio - s qq sttes sclr stte would hve L=1, S=1, J=0 L+S sclr sttes nturl cndidtes for tetrqurks P(940) [u d] u [u d] [ū d] σ(500) Γ=( ) MeV [q s] q [q s] [ q s] Σ(1190) 0 (980), f 0 (980)

30 Δ(1232) L B =0 (u d) u (ud): bd-diqurk S D =1, I D =1, color tetrqurk L T =L B =0 (u d) [ū d] bd-diqurk, good-ntidiqurk S T =1, I T =1 prity P=(-1) L =+ prticle ntiprticle no chrge conjugtion J PC =1 +? d R (u d) R (u d) 2 (1320) L M =1,S M =1 u ū u ū [ū d] [u d] d R (ū d) R (ū d) sttes with definite C cn be obtined

31 , - qq sttes L=1, S=1, J=1 L+S. Nturl for tetrqurks cndidte xil stte J PC =1 ++, I=1 cndidte xil stte J PC =1 +-, I=1 mesonic sttes L=1, S=0, J=1=L+S could mix with tetrqurks 1

32 First L nucleon excittion N 1/2- (1535), N 3/2- (1520) N - (~1530) L B =1 [u d] u [u d] [ū d] tetrqurk I=0, S T =0 L T =1 P =- prticle = ntiprticle C=+ J PC =1 -+ both with I=1 not cndidtes L T =0, I=0 tetrqurk σ(500) huge width L excittion broder or unbound no predictions for the tetrqurk superprtner of N 1/2- (1535), N 3/2- (1520)

33 First L Δ excittion Δ 1/2- (1620), Δ 3/2- (1700) Δ - (~1650) L B =1 (u d) u (u d) [ū d] tetrqurk I=1, S T =1 L T =1 P =- 0 -+, prticle ntiprticle both C re possible: J PC = 1 -+, {2 -+, Tetrqurk cndidtes: only sttes tht do not hve J=L+S s qq - stte P=-(-) L, C=(-) L+S no or sttes (mybe too brod) exotic quntum numbers tetrqurk superprtner of Δ 1/2- (1620), Δ 3/2- (1700)

34 I=0, 1 sttes

35 Extention to Hevy-Light sector DdTB, rxiv: it ws shown tht the LF potentil in the hevy-light sector, even for strongly broken conforml invrince, hs the sme qudrtic form s the one dictted by the conforml lgebr: '( ) = 1, A rbitrry cte 2 A 2 SUSY = {R,R } + µ 2 I, µ 2 =2 S + M 2 [m 1,...m N ]! Q = 1 2 A qurk mss correction

36 Superprteners for sttes with one c qurk predictions betifull greement!

37 Superprteners for sttes with one b qurk predictions

38 Sttes with two hevy qurks Trwinski, Stnislw, Grzek, Brodsky, De Termond, Dosch, PRD90(2104) qudrtic potentil in FF for light qurks liner potentil in IF V=Cr Cornell potentil for hevy qurks I=0, I=1? The LF confinement potentil for systems contining two hevy qurks will be modified. Therefore the extension of superconforml lgebr to such sttes is somewht specultive. However

39 ) 2 Events / 2 MeV/c X(3872) LHCb s = 7 TeV New chrmonium sttes, Z c (3900) chrged stte, I=1!!!!!!! BELLE KEK (2003) (PRL91(2003)) Events/ 5 MeV/c (b) CDF (2004) X(3872) MX = ( ± 0.39) MeV Γ < 1.2 MeV h (c) (d) c LHCb (2012) c c spec. for J PC = ] 0 0 P 1 (3990) (Brnes m & Godfrey, PRD69 (2004)) J/ (GeV/c 2 ) 3 3 P 1 (4290) M(J/ψ π π) [MeV/c 0

40 Chrged Chrmonium stte discovered in 2013 Zc +(3900) B III rxiv: K rxiv: Double b Sector M2( (2S) +), GeV2 Y (4260)! (J/ ) no observed bryonic sttes (yet!)

41 Conclusions SUSY-LFHQCD liner Regee trjectories for mesons, bryons, tetrqurks meson bryon bryon tetrqurk S M = S D S=0

42 Predictions Rλ ℸ : constituent into cluster(2) pentqurks re moleculr sttes ll J PC = 0 ++,1 ++,1 -+ sttes tetrqurk sttes Q Q Q q Q no bryonic bound sttes with 3 hevy qurks SUSY in superconforml QM symmetry properties of hdrons, not to quntum fields no need to introduce new supersymmetric fields or prticles such s squrks or gluinos

43 Thnk you

44 Questions?