1. Hadrons and Nuclei : Chiral Symmetry and Baryons
11/18/2018
Hadrons and Nuclei : Chiral Symmetry and Baryons
Lattice Summer School
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Martin Savage
Summer 2007
University of Washington

2. Why are Baryons Different ?
Baryon masses , MB ~ Lc
Tree-level is fine, quantum loops are the issue M B 4 f 1 p N C
Looking to compute pion and other IR physics,
so re-arrange things to include
mp/MN and q/MN
as perturbations
Jenkins and Manohar 1991

3. Static Source + Perturbative Corrections -- CovariantM = 1 M = 1

4. Get IR correct and UV will follow(1)
Consider complex scalar field : L = j @ Á 2 m P = m Á v + k 2 ( ) v i s a f o u r - e l c t y , 2 = 1 v = ( 1 ; ) i n r e s t f a m

5. Get IR correct and UV will follow(2)k
< m Á v , s e c n d t i a p u b : i p 2 m = ( v + k ) ! 1 O G o d a p r x u n t i l k ! m . h e U V

6. Get IR correct and UV will follow(3)
Field redefintion : consider scalar field Á ( x ) ! e i m v j @ 2 = + L
Removed Classical Trajectory h v = p 2 m Á L i @ + 1 j

7. Heavy Fermions : Generic : HQET , HBcPT , …
Procedure has one more step to it : h v = e i M x P + N H [ ] P = 1 2 ( + v ) L = h v i D H [ 2 M + ] ? 1 ( ) ; O
Systematic covariant expansion in 1/M

8. Why can we do this ? 1 2 M
<

9. Quantum Loops l o p / R » No MN’s !!! i k ¡ m d k ( 2 ¼ ) i v ¢ ¡ m 3Á l o p / R d n k ( 2 ) i v m Á 3
No MN’s !!!

10. Lectures by Claude Bernard
HBcPT (1) = e i 2 M f p + f + O m 2 = Â 1 3 M e V L = f 2 8 T r @ y m q + : ! R
Lectures by Claude Bernard

11. Baryons : Only transformations under vector subgroup are known !!
HBcPT (2)
Baryons : Only transformations under vector subgroup are known !! = 2 ! L R y U ( ; x ) u g l N ! U ; B y @ N ! U y +

12. HBcPT (3) V = 1 2 ¡ » @ + ¢ f ¼ : A i L = N i ¢ D + 2 g S : @ V f S ;= 1 2 @ y + f : A i L = N v i D + 2 g ( ) A S n : @ V
Chiral limit of $g_A$ f S v ; g = 1 2 ( ) [ ] i

13. Integral Tricks : Dim Reg= Z d n q ( 2 ) 1 v + i m a b 8 4 O

14. Integrals : cut-off p m { n o a l y t i c I = Z d q ( 2 ¼ ) 1 v ¢ + i4 q ( 2 ) 1 v + i m 3 j : p m q { n o a l y t i c

15. Including the D MD-MN << Lc must include D in cPT L = ¡ T i ¢ D
Jenkins and Manohar… MD-MN ~ mp ~ p L = T v i D + 2 g ( ) S A n N h : c

16. HBcPT M + k v ! Identified small expansion parameters
Mp/Lc , p/Lc
Power-counting explicit in Lagrangian
Power-counting preserved at loop-level
Some operator coefficients constrained by Lorentz-invariance
Reparameterization Invariance M v + k !

17. Nucleon Mass revisited : Chiral Symmetryq L ( m q ) = N T r [ + ] p u d n M 2 1 4 3 g A 6 f :
One-loop diagram m q + = 1 2 y

18. Nucleon Mass including the Dp = + m 2 1 u d 4 3 g A 6 f N F ( )
" l o à i ! # g N

19. sN-Term and Chiral Expansion
Chiral expansion is worse than that of the nucleon mass itself N = + 1 2 m 3 g A 6 f

20. Nucleon Mass
(Silas Beane)

21. Loop Contribution to Magnetic Moment of the Nucleon (1)S ( q k ) S q
Clebsch’s I C Q 4 g 2 A e f Z d n q ( ) S k v + i m p = ( ) g 2 A M N m 4 f + :

22. Baryon Magnetic MomentsF o r s m a l e n u g h p = ( ) g 2 A M N m 4 f + :

23. Magnetic Moments : Coleman-Glashow Relations= e 4 M N i B F + = p : 2 4 5 N M 7 9 8 n 1 6 3
Works as well as can be expected for SU(3) symmetry

24. Baryon Magnetic Moments (3) Surviving RelationsW o r k i n g t N L O , c l u d e m s p q a b w h y : 6 n + 4 p 3 = 1 5 : 9 N M 7 8

25. S-wave electric dipole
g + p p0 + p (1) c a r i e s l N h a s j = l 1 2 M N 4 p s T = i E + ^ k q P 1 2 3 P - w a v e m p l i t u d s E 1 + ; M
S-wave electric dipole h p j e m A i ! N E + :
Electric field

26. g + p p0 + p (2) + m
> E M
Square root cusp

27. g + p p0 + p (3) E = a m b : Vanishes in the chiral limit+ = a m b 2 :
Vanishes in the chiral limit P t h r e s 1 = j q g N 8 M 2 + p m c :
Slope at threshold is calculable

28. Where is HBcPT valid Mp ~ 0 !! Will fail as Mp -> Lc
Expansion in mp/Lc and not (mp/Lc)2
Convergence will be worse than mesons
Measured not by order-by-order values of bare parameters, but by quality of fits order-by-order

29. Finite Volume (1) k = 2 ¼ L n ( ; )
Lattice calculations are performed in a finite volume with (usually) periodic boundary conditions in the spatial directions.
Momentum states are discrete k = 2 L n ( x ; y z )

30. Finite Volume (2) Assuming the Lc >> L-1 I = Z d q ( 2 ¼ ) 1 v ¢
The counterterms are unchanged, but the IR physics … p-loops….are modified
Meson perturbation theory well explored
Baryons properties at finite volume are more recent
Assume that time-direction of lattice is infinite I = Z d 4 q ( 2 ) 1 v + i m ! L 3 X

31. Modified bessel function
Finite Volume (3) 1 L 3 X q ( 2 + m ) = Z d p 6 j K
Modified bessel function
of the 2nd kind
where we have used X n 3 ( y ) = p e i 2
Poisson Formula
and X n f ( ) = Z d 3 y

32. Periodic Boundary Conditions and Images= s u m o v e r i a g c h

33. Finite Volume (4) : the p-regimeK ( x ) ! r 2 e + O 1 =
For m L >> 1 such sums converge rapidly
Power-counting is the same as infinite-volume
Ali Khan et al
Beane .. includes the D O ( p 4 ) O ( p 3 ) m = 5 4 M e V

34. Finite Volume (5) : Nucleon Magnetic Moment

35. Finite Volume (6) : gA

36. Form of FV Corrections to gA1
GNN (L) = gA ( ) + …
mp f2 L3
mp T >> 1
mp L ~ 1 and << 1
mp L >> 1
(Will Detmold and MJS)
(Silas Beane and MJS)

37. Partially Quenched cPT (1)
Discussed by Claude in the meson sector….will not cover basic approach.
Baryons are a bit trickier to include
Irreps of SU(4|2) or SU(6|3) determined in analogous way to heavy-baryon irreps
First discussed by Labrenz+Sharpe S U ( 2 ) L R ! 4 j

38. QCD, Quenching and Partial- Quenching (2)
(Bernard, Golterman, Sharpe) q v p p q v
QCD
Lie-Groups
Graded Lie-Groups
QQCD q ~ q
PQQCD q v s
Valence
Ghost
Sea

39. Partially Quenched cPT (3)s i f y e d w r t S U ( 2 ) v g N a ( 2 ; 1 ) t ~ s b c 3 :
3 valence quarks
2 valence quarks, 1 sea quark
2 valence quarks, 1 ghost
2 valence quarks, 1 sea quark
2 valence quarks, 1 ghost

40. Partially Quenched cPT (4)B i j k h Q ; a b c ( C 5 ) B i j k = ( ) 1 + T Q ; a i ( x ) b y j = 3 a t , ~ s b c e m d i n B j k u q l y o f h v r .

41. Partially Quenched cPT (5)s c o n t r u e d b v a h g f m N , a ~ t s b , Â

42. Proton Mass in PQQCD with Isospin-Symmetric Sea Quarks
Singlet Axial Coupling
DNp Coupling

43. Partially-Quenched Protons…. Mass Differences

44. Strong Isospin Breaking from Isospin Symmetric Lattices
(S.Beane, K.Orginos, mjs )
Neutron-Proton Mass Difference :
Quark mass differences , md-mu
Electromagnetism
Partially-Quenched Lattice Calculations + Theory
Mn - Mp= MeV
Due to md > mu

45. The END