Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass - PowerPoint PPT Presentation

Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass gap Sreeraj T P The Institute of Mathematical Sciences. 25 July, 2018 Lattice 2018 Work done with : Ramesh Anishetty Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Introduction Attempts to describe Yang mills theory in terms of Gauge invariant Wilson loops. Non-local. Over-complete. We will describe gauge theory in ’dual’ electric loop representation. local complete. Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

The plan of the talk. 1 A quick look at Hamiltonian LGT . 2 Point split lattice - PSlattice. 3 Local gauge invariant states. 4 Path integral in phase space. 5 Weak coupling limit and mass gap. Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Hamiltonian SU(2) Gauge theory on a lattice (Kogut and Susskind, 1976) E a L U ( n , i ) E a R • U ∼ e iA − SU (2) parallel transport operator • E L ≡ lattice analogue of E • E R ≡ − E L parallel transported by U • E 2 R = E 2 L [link constraint] • E L / E R ∈ SU(2) algebra. Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Hamiltonian SU(2) Gauge theory on a lattice continued.. • Hamiltonian is: g 2 H = ˜ 1 � � E a E a + [2 − TrU p ] 2 g 2 2 links plaq Physical states are gauge invariant. Gauss Law Constraints! E a L � � � E a E a E a L ( i ) + E a R ( i ) | ψ phys � = 0 R L E a R i • Gauss law operator generates gauge transformations at each site. • Gauss law says: at each site, incoming electric flux = outgoing electric flux. Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Gauge invariant, local Hilbert space E 2 E a i ∈ su (2) algebra E ¯ E 1 1 Gauss law: E ¯ E 1 + � � 1 + � E 2 + � 2 E ¯ E ¯ 2 = 0 � � � � E 1 + � � E 2 + � � � � � � � � � � E 1 + � � � 1 + � � E 1 + � � 1 + � E ¯ + E ¯ = 0 E 2 + E ¯ E ¯ = 0 E ¯ + E ¯ E 2 = 0 1 2 2 2 = = = j 2 j 2 j ¯ j 1 j 2 1 j ¯ 2 = j 12 1¯ j ¯ 12 = j 1¯ j ¯ 12 = j 1¯ 2 2 j ¯ j 1 j ¯ j 1 1 1 j ¯ j ¯ 2 2 j ¯ 2 (Ramesh Anishetty and H. S. Sharatchandra,PRL,65, 813 (1990)) Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Splitting of point • Split the site into two sites and introduce a new link. • Introduce Link operator and link constraint at the new link. • All sites have 3 links and Gauss law constraint at each site. • Dynamics is much more transparent on the split lattice. (Ramesh Anishetty and T P Sreeraj, PRD, 97, 074511 (2018)) Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

PS-lattice=original lattice • PS lattice reduces to the original lattice by a gauge fixing. 1 2 Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

PS-lattice • Lattice after splitting each site: • plaquette → octagon f e g d n 2 m h 2 n 1 ¯ c 2 3 a b 3 3 1 1 ¯ ¯ 1 1 • 3 possible point splitting schemes at each site → large number of unitarily equivalent Hilbert spaces. Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Schwinger Bosons. • E L , U , E R → a † α ( L ) , a † ; a † α ( R ) α ( L / R ) − Harmonic oscillator doublets! a † a † α ( L ) α ( R ) N L = N R E 2 L = E 2 E a E a R L R L ≡ a † ( L ) σ a R ≡ a † ( R ) σ a E a E a 2 a ( L ) , 2 a ( R ) . � N � E 2 = N 2 + 1 2 � � � � a † a † a † 1 1 2 ( L ) a 1 ( L ) 1 ( R ) 2 ( R ) U = � ˆ � ˆ ( prepotential rep ) − a † 1 ( L ) a 2 ( L ) a 2 ( R ) − a 1 ( R ) N + 1 N + 1 � �� � � �� � U L U R [ Manu Mathur, J.phys A(2005), Phys. Lett. B (2007), Nucl.Phys.B(2007) Ramesh Anishetty, Manu Mathur, Indrakshi. R, JMP(2009),J.Phys(2009),JMP(2010) ] • Under gauge transformations: U → Λ L U Λ † R a ( L ) → Λ L a ( L ) , a ( R ) → Λ R a ( R ) Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Gauge invariant basis with Schwinger Bosons • At a 3-vertex: l 1¯ 1 1 ¯ 1 l 31 l ¯ 13 3 • Normalized gauge invariant states at a 3-vertex: l 1¯ 1 ( a † [¯ l ¯ 13 ( a † [3] ǫ a † [1]) l 31 13 , l 31 � = ( a † [1] ǫ a † [¯ 1] ǫ a † [3]) 1]) | l 1¯ 1 , l ¯ | 0 � ≡ | n 1 , n ¯ 1 , n 3 = m � � ( l 1¯ 1 + l 31 + l ¯ 13 + 1)!( l 1¯ 1 )!( l 31 )!( l 23 )! 1 , n 3 gives the number of harmonic oscillators on the link 1 , ¯ • n 1 , n ¯ 1 , 3. n 1 = l 12 + l 31 n 2 = l 23 + l 12 n 3 = m = l 31 + l 23 (Ramesh Anishetty and T P Sreeraj, PRD, 97, 074511 (2018)) Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

l ¯ l ¯ 13 23 ¯ 2 ¯ a 3 b 1 1 2 l 31 l 32 • Equivalent descriptions based on: 1 l ij satisfying the link condition : l 31 [ a ] + l ¯ 13 [ a ] = n 3 ( ≡ m ) = l 32 [ b ] + l ¯ 23 [ b ] l ij into a link = l ij going out = ⇒ Closed Electric flux loops. 2 n i , m -local quantum numbers satisfying triangle inequalities at each site: | n i − n ¯ i | ≤ m ≤ n i + n ¯ i Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Action of Hamiltonian on the number basis. � ˆ � ˆ • E 2 N i N i i = 2 + 1 diagonal. 2 • TrU p = TrU o changes n i , m at each link along a plaquette by ± 1. ± ± ± = C TrU o ± ± ± ± ± Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Phases N i , e i ˆ φ ] = e i ˆ • We define phase operators satisfying : [ ˆ [ ˆ φ M , e i ˆ χ ] = e i ˆ χ χ φ m n i n ¯ i � � � � e i ˆ � � D e i ˆ χ φ 0 0 � F TrU o | n i , n ¯ i , m � = Tr | n i , n ¯ i , m � e − i ˆ e − i ˆ χ 0 F D φ 0 oct � �� � P =ˆ ˆ L χ ˆ V ˆ L φ � � ( n i + n ¯ i + m + 3)( n i − n ¯ i + m + 1) ( n ¯ i − n i + m + 1)( n ¯ i + n i − m + 1) D = F = 4( m + 1)( n i + 1) 4( m + 1)( n i + 1) Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Path integral in phase space • Path integral is constructed in phase space by usual time slicing and sandwiching eigenbasis of the number and phase basis. • Path integral in phase space is : � ��� � � � �� � � g 2 ′ χ )+ ˜ � i ( n 1 ˙ φ 1 + n 2 ˙ n 2 1 ( s )+ n 2 1 � � φ 2 + m ˙ 2 ( s ) + � 2 − Tr − dt P � 2 g 2 2 s oct oct Z = D φ i D χ e n 1 , n 2 , m n 1 , n 2 , m should satisfy triangle inequality. Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Weak coupling analysis • When g → 0, � n 1 � = � n 2 � = N , � m � = 2 N , N large , φ i , χ small gives � � e i ˆ � � e i ˆ χ � � D φ � 1 � F 0 P = → (1) e − i ˆ e − i ˆ χ 0 1 F D φ � ( n i + n ¯ i + m + 3)( n i − n ¯ i + m + 1) D = ∼ 1 4( m + 1)( n i + 1) � ( n ¯ i − n i + m + 1)( n ¯ i + n i − m + 1) 1 F = ∼ √ 4( m + 1)( n i + 1) 2 N attains the minimum of the magnetic term. • Splitting fields into mean field and fluctuations. n i = N + ˜ n i m = 2 N + ˜ m √ D ∼ o (1) F ∼ o (1 / 2 N ) (2) • Redefine φ i , χ → g φ i , g χ . Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Weak coupling Vacuum • � n 1 � = � n 2 � = N , � m � = 2 N N = ⇒ all electric flux into a site in x direction goes to y direction and vice 2 N versa N N = ⇒ small electric loops. N • Vacuum dominated by small (spatially) electric flux loops containing huge fluxes. (in the unsplit lattice) Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

Fluctuations • Dominant fluctuations: + - - - + + + + - - + + - - + - • sub dominant fluctuations of order 1 N : − − − f ′ + · · · 1 Each flip gives a factor of N . + − √ 2 f + + + Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass

• We now make an expansion in 1 N and g . After a few field redefinitions gives : � 1 � � � �� � m 2 + V ( φ 1 , φ 2 , χ ) 2 − Tr P ≈ 4 N 2 ˜ (3) �� V ( φ 1 , φ 2 , χ ) = g 2 φ 2 − 1 φ 1 + 1 �� 2 � � � (∆ 1 2 ∆ 2 χ − ∆ 2 2 ∆ 1 χ ) 2 + 1 � ( φ 1 + 1 2 ∆ 1 χ ) 2 + ( φ 2 − 1 � 2 ∆ 2 χ ) 2 + χ 2 � 16 N �� φ 2 − 1 φ 1 + 1 � 2 � − (∆ 1 ∆ 2 χ ) 2 � � � � − ∆ 1 2 ∆ 2 χ − ∆ 2 2 ∆ 1 χ + ∆ 1 ∆ 2 χ �� = g 2 � 2 + 1 � � 2 � 2 + χ 2 � � φ ′ 2 1 + φ ′ 2 ∆ 1 φ ′ 2 − ∆ 2 φ ′ ∆ 1 φ ′ 2 − ∆ 2 φ ′ 16 − 1 + ∆ 1 ∆ 2 χ 1 2 N � − (∆ 1 ∆ 2 χ ) 2 (4) Sreeraj T P Weak coupling limit of 2 + 1 , SU (2) lattice gauge theory and mass