A charming trap for soft pions ( Bingwei Long ) ( Sichuan - PowerPoint PPT Presentation

A charming trap for soft pions 龍 炳蔚 ( Bingwei Long )� � 四川大 學 成都 ( Sichuan U., Chengdu, China ) YITP, Kyoto, 11/2016

Λ c (2595) + as an S-wave resonance in πΣ c channel 3/2– ~1 MeV above threshold — extremely shallow ∇ 1/2– Δ Width ∼ 2MeV — narrow π 3/2+ π 1/2+ Strong attraction ( I=0 , L=0 ) between Σ c and a very ππ π soft pion ( Q ~ 20MeV) 1/2+ Pion mass diff. ignored for the moment Λ c Σ c Can a Σ c trap two soft pions? I=0 I=1

(very) Brief intro to Chiral EFT 3-momenta QCD pert. theory Few GeVs Lattice QCD ~ 1 GeV Chiral EFT ~ Few MeVs

Chiral symmetry Approximate symmetry SU(3) L × SU(3) R of QCD Lagrangian / − m f ) q f − 1 � 4 G aµ ν G µ ν L QCD = q f ( iD ¯ a . f = u,d,s, c,b,t Quark masses m f → 0 / q L,l ) − 1 L 0 4 G aµ ν G µ ν � QCD = (¯ q R,l iD / q R,l + ¯ q L,l iD a . l = u,d,s Invariant under ( ) ! ( ) u R u L u L ! ! ! u R ! SU(3) R SU(3) L d R d L d L q L ⌘ 7! d R q R ⌘ 7! s R s L s L s R !

Pions as Nambu-Goldstone bosons Switch to two flavors: u and d However, QCD vacuum (ground state) not invariant under chiral rotations, SU(2) A , the axial part of SU(2) L × SU(2) R 
 ⇒ spontaneous breaking of SU(2) A Pions are Nambu-Goldstone bosons Would be massless if m u,d = 0 Couplings of pions to other particles (including self interactions) proportional to momenta, ∝ Q , or squared mass, ∝ m π 2 


Pion-baryon interactions Pion-baryon interactions constrained by spontaneous broken chiral symmetry: Some examples • Coupling constants may be fixed, e.g. Weinberg-Tomozawa for Σ c i π a ˙ π b − π b ˙ Σ a † � π a � Σ b f 2 π • Coupling constants may NOT be fixed → Low Energy Constants (LEC) � þ i g Σ † ˙ σ · ~ ϵ abc Σ a † ~ ∇ π b Σ c b 0 Σ a π a ˙ π b Σ b f π Σ c axial coupling πΣ c S-wave

Power counting for Q ~ m π Nucleon propagator — 1/Q Two - pion exchanges of nuclear forces Pion propagator — 1/Q 2 Loop integral — Q 4 /(16 π 2 ) ✓ Q ◆ 2 A pion loop brings a suppression factor of 4 π f π Naive dimensional analysis assumed for undetermined LECs 
 ⇒ Minimal number of LECs at a given order

RG inv. constrains PC 3-momenta Cutoff → arbitrary separation between High-engery short and long-range physics states Cutoff Cutoff independence (RG invariance) 
 ⇒ free of modeling short-range physics Low-energy states Modify PC if it violates RG invariance

Λ c (2595) + as an S-wave resonance in πΣ c channel 3/2– 1~2 MeV above threshold — extremely shallow ∇ 1/2– Δ Width ∼ 2MeV — narrow π 3/2+ π 1/2+ Strong attraction ( I=0 , L=0 ) between Σ c and a very ππ π soft pion ( Q ~ 20MeV) 1/2+ Pion mass diff. ignored for the moment Λ c Σ c Can a Σ c trap two soft pions? I=0 I=1

☞ S-wave resonances from a mock-up potential V V 2 A mock-up potential to produce S-wave resonances wave func. E R → r V 1 Resonance ≈ a would-be bound state coupled to continuum Shallow ⇒ tuning V 1 so E R → 0 Narrow 
 ⇒ tuning V 2 , weakly coupled to continuum, so width → 0 Less tuning for higher partial waves, thanks to centrifugal barriers

S -wave resonance poles (Hyodo ’13) 1 Effective range f (0) = − 1 expansion : 2 k 2 − ik a + r k bound state Fixing r and tuning a = 1/r In higher waves, two poles meet at threshold Λ c (2595) + ( Λ c* ) shallow and narrow ⇒ both r and a are Virtual large

Σ c decay width ~ 2MeV, approximated as stable Small pion momenta, Q ~ 20MeV → k 0 = m π + O(k 2 /m π ) δ ~ 1MeV above πΣ c threshold symmetry, crucial!) Ψ coupled to the S wave of πΣ c → time derivative on π (chiral c → π Σ c Λ ? k µ h : O(1) 3 f π √ ⌘ ⇣ π a Ψ + h.c. Σ a † ˙ h Ψ : Λ c * Explicit field of Λ c (2595) +

Counting (very) soft pions π Σ c (BwL ’15) pion prop. ~ 1/ Q 2 baryon prop. ~ 1/( Q 2 /m π ) d 4 l Q 5 (2 π ) 4 ∼ 1 Z 4 π m π nonrelativistic

Counting (very) soft pions π ⇠ m 3 m ⇡ 1 m ⇡ ⇡ f ⇡ Q 2 /m ⇡ f ⇡ f 2 ⇡ Q 2 Σ c m 3 Q 5 m 3 1 1 π π f 2 π Q 2 4 π m π Q 2 Q 2 /m π f 2 π Q 2 m 3 ✏ m π π , ✏ ≡ m 2 ⇡ / 4 ⇡ f 2 ⇡ = 0 . 18 ∼ f 2 π Q 2 Q δ ~ 1MeV Resummation ⟺ that � ∼ Q 2 /m ⇡ ∼ ✏ 2 m ⇡ , Q ~ 20MeV

π πΣ c scattering + + … Σ c 1 ◆ 2 a = h 2 m 2 ✓ 140MeV 1 1 f (0) = π m π − ∆ ∼ − 1 2 k 2 − ik 4 π f 2 328MeV 4MeV a + r π ◆ 2 r = − 4 π f 2 ✓ 328MeV 1 (Hyodo ’13) π ∼ h 2 m 3 140MeV 140MeV r = - 19 fm � ⇒ h = 0.65 π a = - 10 fm p r can be quite large when ∆ ⌧ 4 π f π = 328MeV a single fine-tuning makes both a and r large ∆ − m π → 0 → Chiral symmetry helps Λ c (2595) + be shallow AND narrow (BwL ’15)


 
 
 Can a Σ c attract more pions? very soft π ’s interact w/ other hadrons weakly πΣ c potential is energy-dependent 
 → more complicated than independent-boson systems 
 π h 2 m 2 ⇡ f 2 ⇡ ( E � � ) Σ c Searching 3-body states by finding poles of πΛ c* “scattering amplitude” (or any other correlation func. having same quantum numbers as ) h 0 | ⇡ a Ψ ⇡ a Ψ † | 0 i

πΛ c* scattering ⇠ m 3 m ⇡ 1 m ⇡ = + ⇡ f ⇡ Q 2 /m ⇡ f ⇡ f 2 ⇡ Q 2 Comparable m 3 m 3 ⇡ Q 2 ∼ m 3 Q ∼ ✏ m π Q Q ⇡ ⇡ ⇡ f 2 ⇡ Q 2 4 ⇡ f 2 f 2 ⇡ Q 2 ✏ m ⇡ = + ... + + = + ⇒ Solving ... + = + +

Estimating corrections of m 2 ⇡ /f 2 ⇡ Pion s-wave interaction ∼ ✏ 2 m 3 π f 2 π Q 3 m π /f 2 π Weinberg - Tomozawa (g.s. Λ c+ ) + π is more energetic, Q’ ~ 3m π but still suppressed ✏ 2 m 3 2 ✏ 2 ( Q 0 3 4 ⇡ f π ) 2 , π Λ + c f 2 π Q 3 g.s. Λ c+

Integral equation q : 3-mom. Z l 2 3( q 2 + B ) + 2 8 ⇡ / | r | t ( q ; E , B ) = dl q 2 − E + l 2 + i 0 3 ⇡ Σ l E : total CM energy → t ( l ; E , B ) , e E ≡ 2 m ⇡ E , × √ a − | r | − 1 l 2 − E − i 0 − i 0 2 ( E − l 2 ) + � � − 1 E Λ : Λ c * energy � ✏ h 2 m ⇡ � 1 /a = ✏ h 2 , r = � − − , B ≡ − 2 m ⇡ E Λ when q → ∞ , t ( q ) → 1/ q 2 
 ⇒ integral converges ⇒ cutoff independence 3-body resonances 
 = poles of t ( q ; E , E Λ ) as a function of E

Z l 2 3( q 2 + B ) + 2 8 ⇡ / | r | t ( q ; E , B ) = dl q 2 − E + l 2 + i 0 3 ⇡ Σ l t ( l ; E , B ) Deform the contour so as , × √ a − | r | − 1 l 2 − E − i 0 − i 0 2 ( E − l 2 ) + NOT to cross any singularities of the integrand Instead of l 2 , looking at h ω l ≡ E − l 2 , r ( q 2 − E + l 2 ) � 1 = ( E − ω l − ω q ) � 1 Poles of C Poles of dressed Λ c * prop. C' Branch cut √ l 2 − E = √− ω l , B l - singularities of t ( l ; E , B ) p

Deforming contour C C ⇒ C' C' Solid line: contour in omega plane Thick line: square root cut B Dashed line: cut as a func. of l t ( l ; E , B ) p Cross: poles of dressed prop. Be wary of “standard” procedures (e.g. Peace & Afnan)

3-body resonance pole (BwL ’16) Im e E Re e E − 2 2 4 6 8 e E ≡ 2 m π Er 2 − 4 − 8 − 12 3B pole trajectory as a varies, with | r | -1 as unit | r |/ a = - 4 ~ - 1

Results (BwL ’16) h 2 = 3 (CDF ’11) M Λ ? c � M Λ + c = 305 . 8 MeV , 2 ⇥ 0 . 36 , ⇒ M Σ ( ⇡⇡ Σ c , 1 2 ) � ( M Σ c + 2 m ⇡ ) = ( � 0 . 45 � 0 . 02 i )MeV h 2 = 3 (Chiladze & Falk ’97) M Λ ? c � M Λ + c = 308 . 7 MeV , 2 ⇥ 0 . 30 ⇒ M Σ ( ⇡⇡ Σ c , 1 2 ) � ( M Σ c + 2 m ⇡ ) = (4 . 00 � 5 . 72 i )MeV .

? y Λ + c (2765) 0990700-011 800 600 y Λ + c (2765) Events / 5 MeV The decay of Σ ( ππ Σ c , 1 2 ) into Λ + c π − π + . 400 200 0 Σ c ( Σ ? c ) Σ c Λ c 470 670 770 370 570 M (MeV) Observation of New States Decaying into L 1 c p 2 p 1 (CLEO ’01)

Summary Λ c (2595) + a near-threshold S -wave resonance coupled to πΣ c Strong attraction of very soft pions to Σ c : A extremely rare realization of S-wave resonant interaction with both large a and r Thanks to chiral symmetry, only one fine-tuning needed It helps form a shallow ππΣ c resonance More molecular states with this soft-pion attraction?