National Nuclear Physics Summer School Lectures on Effective Field Theory. Brian Tiburzi. RIKEN BNL Research Center

Transcription

1 2014 National Nuclear Physics Summer School Lectures on Effective Field Theory I. Removing heavy particles II. Removing large scales III. Describing Goldstone bosons IV. Interacting with Goldstone bosons Brian Tiburzi RIKEN BNL Research Center

2 Effective Field Theory IV. Interacting with Goldstone bosons

3 Bottom Up EFTs Effective d.o.f. arise non-perturbatively CMT: quasi-particles QCD: low-lying hadrons End point: breakdown of EFT at higher-scale µ Ω ω Starting point: low-scale EFT Power corrections (ω/ω) n Non-analytic corrections log ω Ω Captures non-analyticities of IR d.o.f. Matching with experiment / non-perturbative calculation Focus: spontaneous symmetry breaking for which EFT is constructed to account for Goldstone modes. QCD chiral symmetry & pions... now add nucleon

4 Nucleon in Chiral Perturbation Theory L N = N (id/ M N ) N Λ χ 1.1 GeV M N =0.94 GeV µ ChPT is a low-energy effective theory p µ = M N v µ + k µ p π,m π Include nucleon as an external flavor source, and describe small energy fluctuations about the nucleon mass k M N Λ χ Account for quark mass dependence, need chiral limit nucleon mass M

5 Nucleon in Chiral Perturbation Theory Digression: chiral limit mass out of nothing! QCD energy-momentum tensor T µν T µ µ = m q ψψ Classically M =0 N( k ) T µν N( k ) = k µk ν M trace N( k ) T µ µ N( k ) = M Trace anomaly (QCD cannot be defined without a scale) M N = T µ µ = β 2g Gµν G µν + m q ψψ N( k ) β 2g Gµν G µν + m q ψψ N( k ) Higgs doesn t have a monopoly over all masses in the universe? = M + Am q + Bm 2 q +... = M + Am 2 π + Bm 4 π +... Investigate: Is the trace of the energy-momentum tensor the divergence of a current?

6 Heavy Nucleon ChPT p µ = Mv µ + k µ Large chiral limit mass phased away: derivative expansion is now valid L = N v iv P + N v µ N v (x) k µ, k µ M Λ χ Combine heavy nucleon limit with chiral perturbation theory: quark mass dependence of nucleon properties, pion-nucleon interactions,... p SU(2) V N = N n i V ij N j Σ = e 2iφ/f Σ SU(2) L SU(2) R LΣR Actually it s unknown to which chiral multiplet(s) the nucleon belongs Assume simple N R RN R N L LN L N Dressed nucleon Ñ R = Σ N L Ñ L = ΣN R πn Exploit arbitrary nature for simplicity ξ = Σ = e iφ/f

7 Heavy Nucleon ChPT Seems complicated: ξ Lξ 2 R LξU Vector subgroup U(L = R = V,ξ) =V ξ V ξv Σ Lξ 2 R ξ 2 LξU LξU ξr = U LξU UξR = LξU Differently dressed nucleon NR = ξ N L NL = ξn R N R U N R NL U N L Dressed differently, chiral components transform the same way Assume simple N R RN R N L LN L N Dressed nucleon Ñ R = Σ N L Ñ L = ΣN R πn Exploit arbitrary nature for simplicity ξ = Σ = e iφ/f

8 Heavy Nucleon ChPT Actually it s unknown to which chiral multiplet(s) the nucleon belongs 1). Within a given chiral multiplet, nucleon field is not unique 2). Invent nucleon field with chiral components transforming the same ξ LξU = UξR 3). Need not know unknown N i U ij N j (meets known N i V ij N j ) 4). Construct heavy nucleon chiral Lagrangian based on symmetry N πn A µ UA µ U

9 Heavy Nucleon Chiral Lagrangian S µ = 0, σ 2 L = N iv DN +2g A N S µ A µ N Two more invariants: (N v µ N)Tr(V µ ) (N S µ N)Tr(A µ ) Axial coupling free parameter in chiral limit (depends upon chiral multiplet) Vector coupling exactly fixed by pattern of chiral symmetry breaking

10 Heavy Nucleon Chiral Lagrangian L = N iv DN +2g A N S µ A µ N Include external vector and axial-vector fields: local chiral transformation Vector and axial-vector pion fields become gauged Singlet couplings can be turned on externally Singlet vector coupling exactly fixed by electric charge assignments

11 Quark Mass Dependence of the Nucleon Now turn on explicit chiral symmetry breaking due to the quark mass L = ψ L s ψ R ψ R s ψ L s LsR s = m q + Dress the scalar source with pions ξ LξU = UξR M ± = 1 2 ξs ξ ± ξ s ξ UM ± U M N = M + σm q +... Leading quark mass dependence L mq = σn M + N + O(m 2 q) Recall M N = N( k ) β 2g Gµν G µν + m q ψψ N( k )

12 Aside: The Pion-Nucleon Sigma Term σ N 1 2M N N( k) m q ψψ N( k) m q = 1 2 (m u + m d ) ψψ = uu + dd Leading-order result σ N = σm q 2M N +... Sigma term relevant for: mass spectrum, strangeness content, quark mass ratios, pion-nucleon scattering, new physics searches,... σ N = m q 2M N M N m q mass spectrum y = N( k) ss N( k) 1 2 N( k) uu + dd N( k) strangeness content quark mass ratio ms 1 (1 y)σ N = m s m q m q 2M N N( k) uu + dd 2ss N( k) End of lecture will be strange

13 Aside: The Pion-Nucleon Sigma Term σ N 1 2M N N( k) m q ψψ N( k) m q = 1 2 (m u + m d ) ψψ = uu + dd Leading-order result σ N = σm q 2M N +... Sigma term relevant for: mass spectrum, strangeness content, quark mass ratios, pion-nucleon scattering, new physics searches,... pion-nucleon scattering Low-Energy Theorem (Cheng-Dashen) 2M N σ N = t =(k k) 2 45(8) MeV [1990 s] 64(7) MeV [2000 s] (experiment coupled with ChPT analysis) D I=0 (ν =0,t=2m 2 π) Born = 2σ N f (large corrections) = 2 σn f 2 (t =2m 2 π)+ R 39(4) MeV [2010 s] (BMW nucleon spectrum from lattice QCD)

14 Anatomy of One-Loop Computations Chiral expansion of nucleon mass M N = M + σm q +... O(p 2 ) Power counting one-loop diagrams O(p) L = N iv DN +2g A N S µ A µ N d 4 p 1 p 2 p = p3 Odd powers! O(p 2 ) L mq = σn M + N Form all one-loop diagrams from leading-order vertices p 2 d 4 p 1 p 2 = p4

15 Anatomy of One-Loop Computations Chiral expansion of nucleon mass M N = M + σm q +... O(p 2 ) Power counting one-loop diagrams O(p) L = N iv DN +2g A N S µ A µ N d 4 v p p p 2 m 2 π =0 Odd powers! O(p 2 ) L mq = σn M + N Form all one-loop diagrams from leading-order vertices p 2 d 4 p 1 p 2 = p4

16 Anatomy of One-Loop Computations Chiral expansion of nucleon mass M N = M + σm q +... O(p 2 ) Power counting one-loop diagrams O(p) L = N iv DN +2g A N S µ A µ N d 4 pp 1 p 2 1 p p = p3 Odd powers! g2 A f 2 d 4 p p σ p σ p 0 (p 2 m 2 π) Form all one-loop diagrams from leading-order vertices =0

17 Anatomy of One-Loop Computations Chiral expansion of nucleon mass M N = M + σm q +... O(p 2 ) Power counting one-loop diagrams O(p) L = N iv DN +2g A N S µ A µ N d 4 pp 1 p 2 1 p p = p3 Odd powers! g2 A f 2 d 4 p p σ p σ p 0 (p 2 m 2 π) Form all one-loop diagrams from leading-order vertices i p v + i = i p 0 + i = i PV + πδ(p 0) forward propagating heavy nucleon Rest frame v µ =(1, 0, 0, 0)

18 Anatomy of One-Loop Computations Chiral expansion of nucleon mass M N = M + σm q +... O(p 2 ) Power counting one-loop diagrams O(p) L = N iv DN +2g A N S µ A µ N d 4 pp 1 p 2 1 p p = p3 Odd powers! Rest frame g2 A f 2 dp d 4 p σ p σ p p 0 (p 2 m 2 π) p σ p σ p 2 + m 2 = dp π v µ =(1, 0, 0, 0) p 2 p 2 + m 2 π = Form all one-loop diagrams from leading-order vertices dp m 2 1 π dp p 2 + m 2 π m 2 π(λ +finite) Λ 3

19 Anatomy of One-Loop Computations Chiral expansion of nucleon mass M N = M + σm q +... O(p 2 ) Power counting one-loop diagrams O(p) L = N iv DN +2g A N S µ A µ N d 4 pp 1 p 2 1 p p = p3 Odd powers! g2 A f 2 dp d 4 p σ p σ p p 0 (p 2 m 2 π) p σ p σ p 2 + m 2 = dp π p 2 p 2 + m 2 π = Form all one-loop diagrams from leading-order vertices dp m 2 1 π dp p 2 + m 2 π m 2 π(λ +finite) Λ 3 Or just use dimensional regularization = m 3 π

20 Chiral Expansion of Nucleon Mass M N = M + A m 2 π 3πg2 A (4πf) 2 m3 π + B m 4 π log µ2 m 2 π + C +... Compilation courtesy of A. Walker-Loud (Chiral Dynamics 2012)

21 Chiral Expansion of Nucleon Properties Can compute chiral corrections to matrix elements of various currents. For example: quark bilinears ψ Γ ψ, four quark operators ψ Γ1 ψ ψ Γ 2 ψ. Isovector axial current N(p ) J + 5µ N(p ) = u 2S µ G A (q 2 )+q µ S qg P (q 2 ) u G A = g A + Am 2 π <r 2 A >= r 2 + Am 2 π G P (q 2 )= g A q 2 m 2 π Isovector EM current log m 2 π + B +... log m 2 π + B <r2 A > 3 + O(m 2 π) <r 2 E >= A log m 2 π + B + O(m π ) µ = µ 0 + Am π + Bm 2 π(log m 2 π + C) <r 2 M >= A 1 m π + B (log m π + C) Form factors and radii G(q 2 )=G(0) 1 6 q2 <r 2 > +... N(p ) J + µ N(p ) = u v µ G E (q 2 )+ ε ijkq j σ k 2M N G M (q 2 ) Experimental numbers <ra 2 > =0.65 fm <re 2 >p n =0.94 fm u

22 + (1232) N(940) Delta Resonances I = 3 2 J P = 3 2 π(140) Low-energy expansion limited by nearest-lying excluded states M M N = 290 MeV strong decays Nπ Higher resonances: Too much momentum available for decays. m π 1 chiral limit, nucleon only New dimensionless parameter m π m π 1 nucleon with explicit deltas δm N g N g g N 2 (4πf) 2 F mπ O(p 3 )

23 Exercises: Write down all strong isospin breaking mass operators up to second order in the quark mass. What effect does isospin breaking in the pion mass have on the nucleon mass? Deduce the behavior of the nucleon mass splitting as a function of the quark masses. In the chiral limit, the isovector axial current is a conserved current. Is there a constraint on the quark isovector axial charge due to the non-renormalization of this current? What about on the nucleon axial charge?

24 Reminder: Asymptotic Expansions Non-analytic quark mass dependence implies asymptotic expansion (but obviously so: zero radius of convergence) Chiral expansion m 2 π/λ 2 χ 0.02 m 2 π/m 2 ρ 0.03 (may even be OK for larger-than-physical pion masses) m 2 π/m 2 σ 0.08 Heavy nucleon expansion M 800 MeV m π /M 0.2 Delta resonance contributions m π / 0.5 /Λ χ 0.3 Higher orders introduce more parameters (low-energy constants) Makes addressing convergence difficult without knowing the chiral limit values of these parameters

25 Heavy Baryon Chiral Perturbation Theory Lowest lying spin-half baryons form an octet of SU(3) V B = 1 2 Σ B VBV 6 Λ Σ + p Σ 1 2 Σ Λ n Ξ Ξ Λ Couple to the Goldstone modes via ξ = Σ = e iφ/f ξ LξU = UξR Free to choose the chiral transformation of baryon octet of the form B UBU A µ UA µ U D µ B = µ B +[V µ,b] O(p) L =Tr Biv DB +2D Tr BS µ {A µ,b} +2F Tr BS µ [A µ,b] Phased away SU(3) chiral limit mass M B (m q = m s = 0)

26 Heavy Baryon Chiral Perturbation Theory Lowest lying spin three-half baryons form a decuplet of SU(3) V M T M B = 270 MeV T ijk V i i V j j V k k T i j k No question about inclusion for three flavors K m K 500 MeV π m π 140 MeV N D, F Σ M Σ 1200 MeV N C M 1230 MeV O(p) L = T µ (iv D ) T µ +2H T µ S AT µ +2C T µ A µ B + BA µ T µ

27 Quark Mass Dependence of the Octet Baryons Now turn on explicit chiral symmetry breaking due to the quark masses L = ψ L s ψ R ψ R s ψ L s LsR s = m +... Dress the scalar source with pions M ± = 1 2 ξs ξ ± ξ s ξ UM ± U O(p 2 ) L m = b D Tr B{M +,B} + b F Tr B[M +,B] + σ Tr BB Tr (M + ) Similar Gell-Mann Okubo constraint on octet baryon masses from tree-level M GMO = M Λ M Σ 2 3 M N 2 3 M Ξ! =0 M GMO /M B 1% One-loop correction predicted in terms of axial couplings C, D, F 4π 4 M GMO = D, F D, F 3(4πf) 2 (D2 3F 2 ) 3 m3 K m 3 η 1 3 m3 π +Decuplet

28 Anatomy of Decuplet Contribution Intermediate-state angular momentum, pion p-wave C2 f 2 d 4 p 2 i p [p 0 + i][(p 0 ) 2 Eπ 2 p + i] 0 = ±E π i p 2 C C E π = p 2 + m 2 π Contour integration C2 p 2 f 2 dp E π (E π + ) C2 f 2 anti-pion on shell (Eπ 2 m 2 de π) 3/2 π m π E π + poles two-body phase space near threshold s m 2 π Divergences Λ de E 1 2 E 2 + E 2 3 E m 2 π E Λ 3 + Λ Λ + 3 log Λ +m 2 πλ + m 2 π log Λ +finite Renormalization condition M N,Σ,Λ,Ξ mq =0 = M B

29 Gell-Mann Okubo Relation to One-Loop M GMO = 4π 3(4πf) 2 π(d 2 3F 2 ) GMO (m 3φ) 16 C2 GMO F(mφ,,µ) δ2 F(m, δ,µ)=(m 2 δ 2 ) m 2 δ δ log 2 m 2 + i δ + δ 2 m 2 + i δ log m2 µ δm2 log m2 µ 2 Scale dependence? Chiral limit? Experiment M GMO /M B 1% BUT: One-loop chiral corrections to the individual masses are LARGE δm N (µ = Λ χ )/M N = 39% δm Λ (µ = Λ χ )/M Λ = 67% δm Σ (µ = Λ χ )/M Σ = 89% δm Ξ (µ = Λ χ )/M Ξ = 98% Expansion stranger with increasing strangeness Heavy baryons m π /M B 0.1 m K /M B 0.5 m η /M B 0.5

30 Confronting SU(3): the strange quark mass SU(3) L SU(3) R m u,m d m s Λ QCD Unless you re exceptionally lucky, the strange quark mass is probably too large for the success of SU(3) baryon chiral expansion... One approach: integrate out the heavy strange quark mass to use an SU(2) theory SU(2) L SU(2) R m u,m d m s Λ QCD For the nucleon (and pion) this is just SU(2) chiral perturbation theory. Done For the nucleon, we treated it as a heavy external flavor source. Nothing stops us from having strangeness in such a source.... SU(2) chiral perturbation theory for strange hadrons Limited predictive power, but ideal for lattice applications

31 Integrating out the strange quark Use the kaon mass to exemplify m 2 K = 4λ f 2 (m q + m s )+...= 1 2 m2 π + M 2 K +...= M 2 K 1+ m2 π 2M 2 K +... M K m K mq =0 Estimate using SU(3) and pion, kaon masses M K =0.486(5) GeV m K 0 =0.497 GeV Consider the SU(3) expansion of the Sigma baryon mass, schematically M Σ = M B + am 2 K + bm 3 K +... Expand out the strange quark contribution M Σ = M B + a M 2 K + a m 2 π + b M 3 K + b M K m 2 π + b 1 M K m 4 π +... Reorganize into SU(2) chiral limit expansion M Σ = M (2) Σ + σ πσ m 2 π + Am 3 π + Bm 4 π log m 2 π + C +...

32 E.g. SU(2) Chiral Perturbation Theory for Hyperons O(p 3 ) cf. SU(3) expansion at physical pion mass δm N (µ = Λ χ )/M N = 39% δm Λ (µ = Λ χ )/M Λ = 67% δm Σ (µ = Λ χ )/M Σ = 89% δm Ξ (µ = Λ χ )/M Ξ = 98% SU(2) chiral expansion m 2 π Λ 2 χ, m 2 π 2M 2 K Heavy baryon expansion m π M (2)

33 Summary IV. Interacting with Goldstone bosons Spontaneous symmetry breaking can be systematically addressed in EFTs EFT construction is bottom down µ M (2) Λ χ M K p, m π In essence effective d.o.f. arise non-perturbatively Goldstone boson dynamics consequence of pattern of spontaneous and explicit symmetry breaking... also true of their interactions with low-lying hadrons E.g. Chiral perturbation theory provides the tool to account for light quark mass dependence of low-energy QCD observables. Size of light quark mass controls efficacy compared with chiral limit mass of included low-lying hadrons