Hadrons and Nuclei : Chiral Symmetry and Baryons 11/18/2018 Hadrons and Nuclei : Chiral Symmetry and Baryons Lattice Summer School Make sure that the toolbars are open Standard Formatting Drawing Martin Savage Summer 2007 University of Washington

Why are Baryons Different ? Baryon masses , MB ~ Lc Tree-level is fine, quantum loops are the issue M B 4 ¼ 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

Static Source + Perturbative Corrections -- Covariant M = 1 M = 1

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

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 µ ¶ G o d a p r x u n t i l k ! m . h e U V

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

Heavy Fermions : Generic : HQET , HBcPT , … Procedure has one more step to it : h v = e i M ¹ x P + N H ¡ [ ] P § = 1 2 ( + v ¹ ° ) L = h v i ¢ D ¡ H [ 2 M + ] ° ¹ ? 1 · ( ) ¾ º ; ¸ O µ ¶ Systematic covariant expansion in 1/M

Why can we do this ? 1 2 M < ¸

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

Lectures by Claude Bernard HBcPT (1) § = e i 2 M f µ ¼ p + ¡ ¶ f ¼ » + O ¡ m 2 = ¤ Â ¢ 1 3 M e V L = f 2 8 T r £ @ ¹ § y ¤ ¡ ¸ m q + : ! R Lectures by Claude Bernard

Baryons : Only transformations under vector subgroup are known !! HBcPT (2) Baryons : Only transformations under vector subgroup are known !! § = » 2 ! L R y U ( ¼ ; x ) u g l N ! U ; B y @ ¹ N ! U y +

HBcPT (3) V = 1 2 ¡ » @ + ¢ f ¼ : A i L = N i ¢ D + 2 g S : @ V f S ; ¹ = 1 2 ¡ » @ y + ¢ f ¼ : A i L = N v i ¢ D + 2 g ( ) A S n : ¹ @ V Chiral limit of $g_A$ f S ¹ v ; º g = 1 2 ( ¡ ) [ ] i ² ¾ ½

Integral Tricks : Dim Reg = Z d n q ( 2 ¼ ) 1 v ¢ + i ² ¡ m a b ¸ 8 ³ ´ 4 O

Integrals : cut-off p m { n o a l y t i c I = Z d q ( 2 ¼ ) 1 v ¢ + i 4 q ( 2 ¼ ) 1 v ¢ + i ² ¡ m » ¤ 3 j µ ¶ : p m q { n o a l y t i c

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

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 ¹ + k !

Nucleon Mass revisited : Chiral Symmetry q L ( m q ) = ¡ ® N T r [ + ] ¯ p u d n M 2 ¼ µ 1 ¶ 4 3 g A 6 f : One-loop diagram m q + = 1 2 ¡ » y ¢

Nucleon Mass including the D p = + m 2 ¼ µ ® 1 ¯ ¶ u ¡ d 4 3 g A 6 f N ¢ F ( ) " l o à i ² ! # ¹ g N ¢

sN-Term and Chiral Expansion Chiral expansion is worse than that of the nucleon mass itself ¾ N = ¡ ® + 1 2 ¯ ¢ m ¼ 3 g A 6 f

Nucleon Mass (Silas Beane)

Loop Contribution to Magnetic Moment of the Nucleon (1) S ¢ ( q ¡ k ) S ¢ q Clebsch’s I » C ¼ Q 4 g 2 A e f Z d n q ( ) S ¢ ¡ k ² v + i m ¹ p = ( ) ¡ g 2 A M N m ¼ 4 f + :

Baryon Magnetic Moments F o r s m a l e n u g h ¼ ¹ p = ( ) ¡ g 2 A M N m ¼ 4 f + :

Magnetic Moments : Coleman-Glashow Relations = e 4 M N ¹ i B ¾ º F ¹ § + = p : 2 4 5 N M 7 9 8 ¤ n ¡ 1 6 3 ¥ Works as well as can be expected for SU(3) symmetry

Baryon Magnetic Moments (3) Surviving Relations W 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 ¹ n + § ¡ 4 p 3 ¤ = ¥ 1 5 : 9 N M 7 8

S-wave electric dipole g + p p0 + p (1) ¼ c a r i e s l ¼ N h a s j = l § 1 2 M N 4 ¼ p s T ¢ ² = i ¾ ³ 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 ¼ j e m ¹ A i ! N ¾ ¢ E + : Electric field

g + p p0 + p (2) ¼ + ¼ m ¼ § > E M Square root cusp

g + p p0 + p (3) E = a m b : Vanishes in the chiral limit + = a m ¼ b 2 : Vanishes in the chiral limit P t h r e s 1 = j q g ¼ N 8 M 2 · + p m c : ¸ Slope at threshold is calculable

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

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 ¼ L n ( x ; y z )

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 ¼ ) 1 v ¢ + i ² ¡ m ! L 3 X

Modified bessel function Finite Volume (3) 1 L 3 X q ( 2 + m ) ® = Z d ¼ ¡ p 6 j K Modified bessel function of the 2nd kind where we have used X n ± 3 ( y ¡ ) = p e i 2 ¼ ¢ Poisson Formula and X n f ( ) = Z d 3 y ± ¡

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

Finite Volume (4) : the p-regime K º ( x ) ! r ¼ 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 ¼ = 5 4 M e V

Finite Volume (5) : Nucleon Magnetic Moment

Finite Volume (6) : gA

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

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

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

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

Partially Quenched cPT (4) B ° i j k » h Q ® ; a ¯ b c ¡ ² ( 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 .

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 ¼ , Â

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

Partially-Quenched Protons…. Mass Differences

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= 2.26+-0.57+-0.42+-0.10 MeV Due to md > mu

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