Bethe Salpeter studies of mesons beyond rainbow-ladder Complutense Unviversity of Madrid

Transcription

1 Bethe Salpeter studies of mesons beyond rainbow-ladder Complutense Unviversity of Madrid Richard Williams C. S. Fischer and RW, Phys. Rev. D , [arxiv: ] C. S. Fischer and RW, Phys. Rev. Lett , [arxiv: ] C. S. Fischer and RW, in preparation 4. November 2009 Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 1 / 42

2 3. The New 2. The Old 4. Results Contents 1. Introduction 5. Next steps and Conclusions Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 2 / 42

3 3. The New 2. The Old 4. Results Contents 1. Introduction 5. Next steps and Conclusions Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 3 / 42

4 Quantum Chromodynamics Z Z[J; ; ] = D[A; ; ] exp Z 1 (D= + m) + F 2 x 4 Zx + A a J a + + Strong interaction described by QCD Gauge theory with gauge group SU(3) non-abelian self-interactions between gauge fields. Degrees of freedom are quarks and gluons Exhibits asymptotic freedom weak-coupling at high momenta! perturbation theory (PT) applicable Confinement Strong coupling for small momenta (no PT) Physical observables colourless bound-states mesons, baryons,... glueballs, exotics. Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 4 / 42

5 Quantum Chromodynamics Need non-perturbative tools lattice QCD effective theories (NJL, PT) functional renormalisation group Dyson-Schwinger equations Why use Dyson-Schwinger equations?? ab initio continuum approach (IR accessible) (necessarily) gauge-fixed! explore confinement mechanisms Bound-states via Bethe-Salpeter/Faddeev equations! complementary to other approaches DSE requires truncation errors difficult to quantify and control Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 5 / 42

6 Dyson-Schwinger equations Z S D' ' + J exp Z S + J' = 0 Derive from observation that integral of total derivative vanishes.! Quantum analogue of classical equations of motion. n o i = ^A ; ^ ; ; ^ : : : superfield [ i] effective action S[ i] classical action Provides (infinite) set of coupled integral equations for the Euclidean Green s functions. [ i] i S[ i] i " 2 1 i! i + i j j # = 0 : [ F. J. Dyson, Phys. Rev. 75 (1949) ] [ J. S. Schwinger, Proc. Nat. Acad. Sci. 37 (1951) 452. ] [ R. Alkofer, M. Q. Huber and K. Schwenzer, Comput. Phys. Commun. 180 (2009) 965 [arxiv: [hep-th]] ] [ C. S. Fischer and J. M. Pawlowski, Phys. Rev. D 75 (2007) [arxiv:hep-th/ ] ] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 6 / 42

7 Dyson-Schwinger equations quark propagator 1 = 1 + S 1 (p; ) = Z 2 2 ; 2 S 1 0 (p; ) + (p; ; ) (p; ; ) = C F g 2 Z 1F 2 ; 2 Z q i S (q; ) D (k; ) (p; k; ) S(p; ) quark propagator (p; ; ) quark self-energy D (p; ) gluon propagator (k; p; ) quark-gluon vertex. Z 1F ; Z 2 renormalisation constants. S 1 (p; ) = ip= A p 2 ; 2 + B p 2 ; 2 D (p; ) = p p Z(p 2 ; 2 ) p 2 p 2 Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 7 / 42

8 Dyson-Schwinger equations quark propagator Truncation necessary gluon, quark-gluon vertex. Weak-coupling expansion reproduces all diagrams of perturbation theory! m( 2 ) = 0 =) B p 2 ; 2 =A p 2 ; 2 = 0! Dynamical mass generation is non-perturbative strange M(p 2,µ 2 ) [GeV] up/down 10-4 chiral p 2 [GeV 2 ] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 8 / 42

9 Dyson-Schwinger equations gluon propagator 1 = = D (p; ) = p p Z(p 2 ; 2 ) p 2 p 2 Quark loops neglected Faddeev-Popov! ghosts (and ghost DSE) Landau gauge truncating is simpler Solvable and good agreement with Lattice! IR scaling/decoupling a moot point irrelevant for mesons. Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 9 / 42

10 Dyson-Schwinger equations quark-gluon vertex = (k; p; ) = 4X i=1 i (k; p) = f1; p= ; k= ; [k= ; p= ]g f (1) i + f (2) i k + f (3) i p i (k; p) f i (j) (k 2 ; p 2 ; k p; 2 ) scalar dressing functions Twelve covariants in total. Non-trivial momentum dependence come back to this later! tree level, (k; p; ) = or: f (j) i = 1 if i = j = 1 0 otherwise Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 10 / 42

11 Bethe-Salpeter equations connect gauge-dependent Green s functions to physical observables describe colourless bound-states in terms of quarks and gluons appear in quark 4-pt function, G or in amputated connected part T G = G 0 + G 0 T G 0 where G 0 is a product of 2 dressed quark propagators. Relate T matrix to 2-quark scattering kernel K by Dyson sum T = K + KG 0 K + KG 0 KG 0 K + In integral notation (Dyson equation): T = K (1 + G 0 T ) Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 11 / 42

12 Bethe-Salpeter equations Introduce bound state amplitude as residues of scattering matrix: T P 2!M 2! N P 2 + M 2 insert into Dyson eq and compare residues gives bethe-salpeter equation at pole P 2 = M 2. normalisation from T 0 = T T 1 0 T N d dp T 2 1 = 1 =) Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 12 / 42

13 Bethe-Salpeter equations Amplitude/Wavefunction = () (q; P ) = X i i (q; P )T () i (q; P ) Relativistic covariants [ C.H.Llewellyn-Smith, Phys. Lett. B 28 (1968) 335 ] Component T 1 : 1 T 2 : ip= T 3 : ip= T 4 : [p= ; P= ] J P C = 0 +, 1 ++, 1 + : prefixed by 5. Component T 1 : it T 2 : T P= T 3 : T p= + p T 1 T 4 : it [P= ; p= ] + 2ip T P= T 5 : p T 1 T 6 : ip T P= T 7 : ip T p= T 8 : p T [P= ; p= ] So far: said nothing about the truncation. Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 13 / 42

14 Axial-vector Ward-Takahashi identity + = 5 ( p ) + (p + ) 5 = Z K ; (p; q; P ) 5 S ( q ) + S (q + ) 5 axwti connects quark self-energy and the bethe-salpeter kernel truncation must consider this important for chiral properties of pseudoscalars - Gell-Mann Oakes-Renner / massless pion in the chiral limit! difficult and intricate to implement in practice. m 2 f ' 2m q hqqi Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 14 / 42

15 3. The New 2. The Old 4. Results Contents 1. Introduction 5. Next steps and Conclusions Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 15 / 42

16 Rainbow-ladder truncation = = K ; (p; q; P ) = Z 2 2 g 2 i (k; p; ) = iz2 i 2 AC 2 (i ) D (q) (i ) BD One-gluon exchange Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 16 / 42

17 Rainbow-ladder truncation = 1 q 2 = K ; (p; q; P ) = Z 2 2 g 2 i (k; p; ) = iz2 i 2 AC 2 (i ) D (q) (i ) BD One-gluon exchange Also: dress vertex with function 1 q 2 Reduces twelve components of k 2 ; p 2 ; q 2 to one with q 2 dependence. 1 q 2 allows part of vertex-dressing to be included. Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 16 / 42

18 Rainbow-ladder truncation Practical application How do we apply this truncation Phenomenology dictates: feature DSB chiral condensate Other considerations: perturbative limit,... other inputs calculable from QCD Interaction must provide enough interaction strength for DSB Munczek Nemirowksy Maris Tandy effective interaction Watson interaction Soft-Divergent model [ H. J. Munczek and A. M. Nemirovsky, Phys. Rev. D 28 (1983) 181. ] [ P. Maris and P. C. Tandy, Phys. Rev. C 60, (1999) ] [ R. Alkofer, P. Watson and H. Weigel, Phys. Rev. D 65 (2002) ] [ R. Alkofer, C. S. Fischer and R. Williams, Eur. Phys. J. A 38 (2008) 53 ] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 17 / 42

19 Rainbow-ladder truncation Practical application Whole host of things are calculable Bound-state mass Electroweak decay constants Electromagnetic form factors pion charge radius magnetic moments transition form factors (hadronic decays) After fixing phenomenology in Meson sector, apply to three-body systems: Nucleon/Delta from quark-diquark model full-three body equation [ P. Maris, P. C. Tandy, PRC 60 (1999) ] [ P. Maris and P. C. Tandy, Phys. Rev. C 62 (2000) ] [ P. Maris and P. C. Tandy, Phys. Rev. C 65 (2002) ] [ D. Nicmorus, G. Eichmann, A. Krassnigg and R. Alkofer, Phys. Rev. D 80 (2009) ] [ G. Eichmann, PhD thesis (KFU Graz) [arxiv: ] ] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 18 / 42

20 3. The New 2. The Old 4. Results Contents 1. Introduction 5. Next steps and Conclusions Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 19 / 42

21 Beyond Rainbow-Ladder = Symmetry preserving truncation: obtained by cutting internal quark-lines in the quark Dyson-Schwinger equation Perform dressed skeleton expansion to perform truncation IR power counting, 1=N c expansion yields dominant contributions. [ H. J. Munczek, Phys. Rev. D 52 (1995) 4736 ] [ A. Bender, C. D. Roberts, L. von Smekal, Phys. Rev. Lett. B 380 (1996) 7 ] [ R. Alkofer, C. S. Fischer and F. J. Llanes-Estrada, Mod. Phys. Lett. A 23 (2008) 1105 ] [ R. Alkofer, C. S. Fischer, F. J. Llanes-Estrada and K. Schwenzer, Annals Phys. 324 (2009) 106 ] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 20 / 42

22 Beyond Rainbow-Ladder: what has been done? = + Nc 2 1 2N c + Trivialization of gluon momentum Generally, extreme approximations have been made 1 in both the quark DSE and vertex DSE 2 in either the quark DSE or vertex DSE Dominant vertex contribution neglected/untreatable in (1) meson amplitude has p = 0 fixed. Meson states unbound (e.g. scalar, axial-vector). D (q) / q q (4) q 2 (q) Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 21 / 42

23 Beyond Rainbow-Ladder: what has been done? = 1 2N c + 1. Munczek Nemirovsky [ M. S. Bhagwat et al, PRC 70, (2004) ] 2. Munczek Nemirovsky + prefactor [ M. S. Bhagwat et al, PRC 70, (2004) ] 3. Hybrid [ P. Watson, W. Cassing and P. C. Tandy, Few Body Syst. 35 (2004) 129 ] rainbow-ladder abelian correction change m m m m m m Comparing model (1) to (3) we see a sign change. Consequence of zero momentum exchange? qualitative statements from (1,2) dangerous Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 22 / 42

24 Beyond Rainbow-Ladder: what must be done? To investigate dominant contributions we need: a gluon with non-trivial momentum dependence appropriate internal vertex dressings Tackle two-loop integrals over NP propagators and vertices. Challenging. 1 (q 2 )Z(q 2 ) = D! 2 q4 e q2 =! 2 First study: compare to existing literature employ model gluon: provides DSB UV suppressed (don t worry about renormalisation) no trivialization of the gluon momentum! study qualitative impact of vertex corrections! framework generalises to realistic input. Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 23 / 42

25 Beyond Rainbow-Ladder: what must be done? What do we gain? Truncation does not restrict what mesons we can study: a 1 b Quark-gluon vertex: all twelve components full momentum dependence (k 2 ; p 2 ; k p)! imparts quark-mass dependence on interaction. Different interactions: vector vector vector scalar different degrees of attraction/repulsion in different meson channels. Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 24 / 42

26 Beyond Rainbow-Ladder: what must be done? Technical challenges? Auxiliary normalisation condition: (Leon Cutkosky) momentum depdendent kernel complicates evaluation Im p 2 GeV Bound state in Eucl. space, P 0 = im quark and vertex momenta (q P=2) 2 : Re p2 GeV 2 must be evaluated for complex momenta region bounded by parabola dependent upon meson mass. Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 25 / 42

27 Normalisation: Leon Cutkosky 2 tr R k " i S R # j S + [ i q ]srktu;rs[j] ut ij # usual approach to determining normalisation condition lines quark propagator. dashed lines quark propagator fixed wrt. derivative. wiggles gluon propagator. [ R. E. Cutkosky and M. Leon, Phys. Rev. 135 (1964) B1445. ] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 26 / 42

28 Normalisation: Nakanishi d ln() 1 Z = tr dp 2 k S S d ln() " # 1 dp 2 = Where is the eigenvalue obtained via = K. alternative hidden in the literature considerably simpler valid for all truncations first time applied beyond rainbow-ladder [ N. Nakanishi, Phys. Rev. 138, B1182 (1965) ] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 27 / 42

29 3. The New 2. The Old 4. Results Contents 1. Introduction 5. Next steps and Conclusions Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 28 / 42

30 Results: Rainbow-Ladder benchmark Calculate bound-state masses with bare vertex: = 1 = 1 + = Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 29 / 42

31 Results: Rainbow-Ladder benchmark 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1:0 1:1 1:2 a1 b1 0:138 0:645 0:758 0:926 0:912 Mass [GeV] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 30 / 42

32 Results: non-abelian corrections Consider dominant non-abelian corrections to vertex = + 1 = 1 + = + + Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 31 / 42

33 Results: non-abelian corrections 0:0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1:0 1:1 1:2 1:3 Repulsive a1 b1 0:142 0:884 0:881 1:056 0:973 Mass [GeV] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 32 / 42

34 Results: Abelian corrections Consider sub-leading Abelian corrections to vertex = + 1 = 1 + = Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 33 / 42

35 Results: Abelian corrections 0:0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1:0 1:1 1:2 1:3 Attractive Watson/Cassing/Tandy a1 b1? 0:137 0:602 0:734 0:889 0:915 Mass [GeV] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 34 / 42

36 Results: non-abelian and Abelian corrections Put them all-together! = = 1 + = Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 35 / 42

37 Results: non-abelian and Abelian corrections 0:0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1:0 1:1 1:2 1:3 Repulsive a1 (essentially the same as RL + NA) b1 0:142 0:884 0:881 1:056 0:973 Mass [GeV] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 36 / 42

38 Unquenching = = (...) N, = YM 1 1 = YM In addition to non-resonant diagrams resonant contributions from, e.g. pion exchange! approximate and include (Ignore contribution from sub-leading Abelian diagrams) Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 37 / 42

39 Results: non-abelian and unquenching effects 0:0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1:0 1:1 1:2 1:3 Pion corrections attractive Abelian diagram irrelevant a1 b1 RL RL+NA RL+NA+PI 0:138 0:805 1:040 0:820 0:941 Mass [GeV] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 38 / 42

40 Results: non-abelian and unquenching effects Implicit mass-dependence of vertex corrections unquenching: shifts towards corrected lattice results m ρ [GeV] CP-PACS + Adelaide RL RL + non-abelian m π [GeV] [ C. R. Allton, W. Armour, D. B. Leinweber, A. W. Thomas and R. D. Young, Phys. Lett. B 628 (2005) 125. ] Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 39 / 42

41 3. The New 2. The Old 4. Results Contents 1. Introduction 5. Next steps and Conclusions Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 40 / 42

42 Next steps Pion electromagnetic form-factor Beyond Impulse-Approximation Consistent with BSE truncation must satisfy current conservation Dressed quark-photon vertex Explicitly three-loop - (implicitly eight) include inputs from gluon DSE, internal quark-gluon vertex, dressed three-gluon vertex.! examine meson spectrum Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 41 / 42

43 Conclusions Summary Reached an era where rainbow-ladder is no longer the only feasible truncation first study no trivialization of momenta: leading non-abelian corrections sub-leading Abelian corrections also hadronic resonance contributions full calculation: solve the full system of equations without overt simplifications Not just a toy-model foundation on which to build future ab initio bound-state studies! just change the inputs to something realistic! Found that sub-leading Abelian corrections are further suppressed dynamically Dominant non-abelian corrections are generally repulsive take everything with a pinch of salt: wait until realistic study completed Richard Williams, TU Darmstadt Bethe Salpeter studies of mesons beyond rainbow-ladder 42 / 42