Λ QCD and Light Quarks Contents Symmetries of the QCD Lagrangian Chiral Symmetry and Its Breaking Parity and Handedness Parity Doubling Explicit Chira

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1 Lecture 5 QCD Symmetries & Their Breaking From Quarks to Hadrons Adnan Bashir, IFM, UMSNH, Mexico August 2013 Hermosillo Sonora

2 Λ QCD and Light Quarks Contents Symmetries of the QCD Lagrangian Chiral Symmetry and Its Breaking Parity and Handedness Parity Doubling Explicit Chiral Symmetry Breaking Gell-Mann Mann-Oakes Oakes-Renner Formula Gell-Mann Mann-Okubo Mass Formula On Quark Masses Charting out The Q 2 Evolution

3 Λ QCD and Light Quarks We analyze the expression for the running QCD coupling: As B is positive, when Q 2 decreases from Q 2 >> Q 2, α s (Q 2 ) increases. We define Q =Λ QCD, so that α s (Q 2 ) ->. Then we expect perturbation theory to break down much above that scale.

4 Λ QCD and Light Quarks Let us assume that perturbation theory breaks down at:

5 Λ QCD and Light Quarks Let us look at the quark mass terms. Light quark masses: Heavy quark masses:

6 Symmetries of the QCD Lagrangian Look at the QCD Lagrangian with u, d and s quarks: The mass term is now:

7 Symmetries of the QCD Lagrangian U(1) Vector Symmetry: This implies conservation of individual flavors, of baryon number and electromagnetic charge in strong interactions. SU(2) Vector Symmetry: It is exactly conserved for equal u and d masses. It implies that the hadrons made out of u and d quarks within a given multiplet should have equal masses. SU(3) Vector Symmetry: It is exactly conserved for equal u, d and s masses. It implies that the hadrons made out of u, d & s quarks within a given multiplet should have equal masses. The SU(3) has three subgroups of SU(2) nature. Axial symmetries for massless quarks can also be defined

8 Symmetries of the QCD Lagrangian Global Chiral SU(2) R X SU(2) L Symmetry: We can work either with the symmetry transformations SU(2) V X SU(2) A or SU(2) L X SU(2) R. Massless Lagrangian: SU(2) R X SU(2) L : The Lagrangian remains invariant. It is chiral symmetry involving only up and down quarks. S quarks can readily be included to study SU(3) R X SU(3) L chiral symmetry. What is chirality?

9 Chiral Symmetry and Its Breaking

10 Chiral Symmetry and Its Breaking How pions, rhos and other mesons transform under chiral transformations? Let us consider combinations of quark fields, which carry the quantum numbers of the mesons under consideration: The vector sign indicates the iso vector nature of the particle. The μ index is the Lorentz index (vector particle).

11 Chiral Symmetry and Its Breaking See how a pion transforms under SU(2) V transformations. The rho:

12 Chiral Symmetry and Its Breaking See how mesons transforms under SU(2) A transformations. The fermions: The mesons:

13 Parity and Handedness Recall: Parity of a spinor wave functions is determined from: Dirac spinors tansform under parity as:

14 Parity and Handedness Left and right handed spinors do not have specific parity: Hpwever, following combinations have specific parity:

15 Recall that axial symmetry is a symmetry of the massless QCD Hamiltonian. This should imply that the states which can be rotated into each other by this symmetry operation should have the same eigenvalues, i,e., the same masses. The linear combinations of Parity Doubling of the left and right handed charge operators commute with the massless QCD Hamiltonian. They have opposite parity. Thus for any state of positive parity, one would expect the existence of a degenerate state of negative parity (parity doubling).

16 Parity Doubling Degeneracy of parity partners can also be shown as follows: Let us look at parity partners:

17 Parity Doubling

18 Explicit Chiral Symmetry Breaking Global Chiral SU(2) V X SU(2) A Symmetry: Vector and axial vector currents: Explicit transformations & currents are: Mass term breaks Chiral or axial symmetry: As long as masses are small as compared to a relevant mass scale, the symmetry is almost (partially) conserved. u and d masses are 5-10 MeV which is much smaller than Λ QCD.

19 Explicit Chiral Symmetry Breaking

20 Gell-Mann-Oakes-Renner Formula Let us take m u m d, i.e, let us break the isospin symmetry Let us define: Then:

21 Gell-Mann-Oakes-Renner Formula We would like to relate observable quantities like pion decay constant and pion mass to QCD quantities like the current masses of the quarks and the condensates. We start from the general expression for SU(3): Let us look at a=1:

22 Note that: Gell-Mann-Oakes-Renner Formula Thus: Compare this with current algebra formula which is derived using anti-commutation relations between the quark fields: We thus have:

23 Gell-Mann-Oakes-Renner Formula To calculate the left hand side, we insert: Note that for the case of spontaneous symmetry breaking, complete set of states is exhausted by the Goldstone Bosons alone (nothing else contributes). In case of SU(3), these are pions, kaons and eta. Normalization confirmed:

24 Gell-Mann-Oakes-Renner Formula We can now evaluate the LHS at x=0: Recall that: Thus:

25 Gell-Mann-Oakes-Renner Formula Similarly: Thus:

26 Gell-Mann-Oakes-Renner Formula Thus we finally arrive at: It helps estimate the quark-anti anti-quark condensate:

27 Gell-Mann-Oakes-Renner Formula

28 Gell-Mann-Okubo Mass Formula In general: Employing: And using current algebra:

29 Gell-Mann-Okubo Mass Formula Let us assume the relations which are strictly valid only in the equal mass limit: It implies Gell-Mann Mann-Okubo mass formula: Use phenomenological values: Good prediction:

30 On Quark Masses We can also obtain quark mass ratios: This implies that the strange quark mass is much larger than the up and down quark masses. This relation is approximately satisfied if:

31 What Next? The static properties of low lying hadrons can be well described by symmetry principles, their breaking and the low energy theorems. Low energy effective QCD Models, SDES and lattice describe the properties of light hadrons well. Form factors of hadrons provide a modern testing ground to study QCD from first principles and see if it can enable us to study the transition region where hadrons go from a non perturbative description to them being made of the valence quarks alone. Thus is an active field of current experimental and theoretical research.

32 Charting out the Q 2 Evolution Observing the transition of the hadron from a sea of quarks and gluons to the one with valence quarks alone is an experimental and theoretical challenge.

33 The Charting out the Q 2 Evolution transition form factor: H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez and P.C. Tandy, Phys. Rev. C82, (065202:1-11) 11) CELLO H.J. Behrend et.al., Z. Phys C (1991) GeV 2 CLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998) GeV 2 BaBar R. Aubert et. al., Phys. Rev. D (2009) GeV 2 Belle S. Uehara et. al., arxiv: [hep-ex] (2012) GeV 2

34 The Charting out the Q 2 Evolution transition form factor: CELLO H.J. Behrend et.al., Z. Phys C (1991) GeV 2 CLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998) GeV 2 BaBar R. Aubert et. al., Phys. Rev. D (2009) GeV 2 Belle S. Uehara et. al., arxiv: [hep-ex] (2012) GeV 2

35 The Charting out the Q 2 Evolution transition form factor: The leading twist pqdc calculation: G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22, 2157 (1980).

36 Charting out the Q 2 Evolution The transition form factor: Belle II will have 40 times more luminosity. Vladimir Savinov: 5 th Workshop of the APS Topical Group on Hadronic Physics Precise measurements at large Q 2 will provide a stringent constraint on the pattern of chiral symmetry breaking.