2007/08/10 @ Kinki Univ. Two-dimensional N = (2 , 2) super Yang-Mills theory on computer Hiroshi Suzuki (RIKEN, Theor. Phys. Lab.) arXiv:0706.1392 [hep-lat] 1
It will be very exciting if non-perturbative questions in SUSY gauge theories can be studied numerically at one’s will ! • spontaneous SUSY breaking • string/gauge correspondence • test of various “solutions” (e.g., Seiberg-Witten) SUSY vs lattice ! { Q, Q † } ∼ P SUSY restores only in the continuum limit ! 2
Present status: • For 4d N = 1 SYM (gaugino condensation, degenerate vacua, Veneziano-Yankielowicz effective action, etc.), nu- merically promising formulation exists • Even in this “simplest realistic” model, no conclusive evidence of SUSY has been observed • Investigation of low-dimensional SUSY gauge theories (simpler UV structure) would thus be useful to test var- ious ideas • Kaplan et. al., Sugino, Catterall, Sapporo group. . . • SUSY QM (16 SUSY charges!) ⇐ Takeuchi-kun 3
In this work, we carry out a (very preliminary) Monte Carlo study of Sugino’s lattice formulation of 2d N = (2 , 2) SYM (4 SUSY charges) F. Sugino, JHEP 03 (2004) 067 [hep-lat/0401017] 4
Two-dimensional square lattice (size L ) x ∈ a Z 2 | 0 ≤ x µ < L � � Λ = The lattice action � � 1 S = Qa 2 � O 1 ( x ) + O 2 ( x ) + O 3 ( x ) + a 4 g 2 tr { χ ( x ) H ( x ) } , x ∈ Λ where 1 � 1 � O 1 ( x ) = a 4 g 2 tr 4 η ( x )[ φ ( x ) , φ ( x )] 1 � � − iχ ( x )ˆ O 2 ( x ) = a 4 g 2 tr Φ TL ( x ) 1 1 µ ) U ( x, µ ) − 1 � � � O 3 ( x ) = a 4 g 2 tr i ψ µ ( x ) φ ( x ) − U ( x, µ ) φ ( x + a ˆ µ =0 5
A lattice counterpart of the BRST-like transformation Q QU ( x, µ ) = iψ µ ( x ) U ( x, µ ) µ ) U ( x, µ ) − 1 � � Qψ µ ( x ) = iψ µ ( x ) ψ µ ( x ) − i φ ( x ) − U ( x, µ ) φ ( x + a ˆ Qφ ( x ) = 0 Qχ ( x ) = H ( x ) QH ( x ) = [ φ ( x ) , χ ( x )] Qφ ( x ) = η ( x ) Qη ( x ) = [ φ ( x ) , φ ( x )] Q 2 = 0 on gauge invariant quantities From this nilpotency, the lattice action is manifestly invari- ant under one of four super-transformations, Q . 6
More explicitly 3 6 � 2 1 H ( x ) − 1 � S = a 2 � 2 i ˆ � � L B i ( x ) + L F i ( x ) + a 4 g 2 tr Φ TL ( x ) x ∈ Λ i =1 i =1 where 1 � 1 � 4[ φ ( x ) , φ ( x )] 2 L B 1 ( x ) = a 4 g 2 tr 1 � 1 � Φ TL ( x ) 2 ˆ L B 2 ( x ) = a 4 g 2 tr 4 1 1 µ ) U ( x, µ ) − 1 � � � L B 3 ( x ) = a 4 g 2 tr φ ( x ) − U ( x, µ ) φ ( x + a ˆ µ =0 � µ ) U ( x, µ ) − 1 � × φ ( x ) − U ( x, µ ) φ ( x + a ˆ 7
and 1 − 1 � � L F 1 ( x ) = a 4 g 2 tr 4 η ( x )[ φ ( x ) , η ( x )] 1 L F 2 ( x ) = a 4 g 2 tr {− χ ( x )[ φ ( x ) , χ ( x )] } 1 0) U ( x, 0) − 1 �� � � φ ( x ) + U ( x, 0) φ ( x + a ˆ L F 3 ( x ) = a 4 g 2 tr − ψ 0 ( x ) ψ 0 ( x ) 1 � � 1) U ( x, 1) − 1 �� φ ( x ) + U ( x, 1) φ ( x + a ˆ L F 4 ( x ) = a 4 g 2 tr − ψ 1 ( x ) ψ 1 ( x ) 1 � � iχ ( x ) Q ˆ L F 5 ( x ) = a 4 g 2 tr Φ( x ) 1 1 µ ) U ( x, µ ) − 1 � � � L F 6 ( x ) = a 4 g 2 tr − i ψ µ ( x ) η ( x ) − U ( x, µ ) η ( x + a ˆ µ =0 8
Advantage of this formulation • Q -invariance (a part of the supersymmetry) is manifest even with finite lattice spacings and volume (probably, so far the unique formulation?) • global U(1) R symmetry (this is a chiral symmetry!) ψ µ ( x ) → e iα ψ µ ( x ) U ( x, µ ) → U ( x, µ ) φ ( x ) → e 2 iα φ ( x ) χ ( x ) → e − iα χ ( x ) H ( x ) → H ( x ) φ ( x ) → e − 2 iα φ ( x ) η ( x ) → e − iα η ( x ) is also manifest 9
Possible disadvantage of the formulation • The pfaffian Pf { iD } resulting from the integration of fermionic variables is generally a complex number (lattice artifact) • would imply the sign (or phase) problem in Monte Carlo simulation cf. H.S. and Taniguchi, JHEP 10 (2005) 082 [hep-lat/0507019] 10
Continuum limit: a → 0 , while g and L are kept fixed It can be argued that the full SUSY of the 1PI effective action for elementary fields is restored in this limit • Power counting • scalar mass terms are the only source of SUSY breaking ⇐ super-renormalizability • exact Q -invariance forbids the mass terms 11
Monte Carlo study ( SU (2) only) 12
For SUSY, quantum effect of fermions is vital ! Quenched approximation ( S B bosonic action) � d µ B O e − S B �O� = � d µ B e − S B is meaningless, though it provides a useful standard Here we adopt the re-weighting method � d µ O e − S = �O Pf { iD }� � �O� � = � � Pf { iD }� d µ e − S (potential overlap problem) 13
We developed a hybrid Monte Carlo algorithm code for the action S B by using a C++ library, FermiQCD/MDP For each configuration, we compute the inverse (i.e., fermion propagator) and the determinant of the lattice Dirac op- erator iD by using the LU decomposition Expressing the determinant of the Dirac operator as det { iD } = re iθ , − π < θ ≤ π (the complex phase is lattice artifact) we define Pf { iD } = √ re iθ/ 2 , ∵ (Pf { iD } ) 2 = det { iD } However, with this prescription, the sign may be wrong 14
0.12 β = 4.0 β = 16.0 0.1 0.08 0.06 0.04 0.02 0 -4 -3 -2 -1 0 1 2 3 4 θ To estimate the systematic error introduced with this, we compute also the phase-quenched average � phase-quenched = �O | Pf { iD }|� � �O� �| Pf { iD }|� 15
Parameters in our Monte Carlo study ( β = 2 N c / ( a 2 g 2 ) ) N 8 7 6 5 4 16 . 0 12 . 25 9 . 0 6 . 25 4 . 0 β N conf 1000 10000 10000 10000 10000 0 . 5 0 . 571428 0 . 666666 0 . 8 1 . 0 ag This sequence corresponds to the fixed physical lattice size Lg = 4 . 0 For each value of β , we stored 1000 – 10000 independent configurations extracted from 10 6 trajectories of the molec- ular dynamics Statistical error is estimated by the jackknife analysis (The constant ǫ for the admissibility is fixed to be ǫ = 2 . 6 ) 16
One-point SUSY Ward-Takahashi identities 17
Since the action is Q -exact, we have � � S � � = 0 , or 3 6 � 2 � � � � 1 H ( x ) − 1 � 2 i ˆ � � � �L B i ( x ) � � + � �L F i ( x ) � � + tr Φ TL ( x ) = 0 a 4 g 2 i =1 i =1 but 6 c − 1) 1 � = − 2( N 2 � � �L F i ( x ) � a 2 i =1 and � 2 � � � � 1 H ( x ) − 1 = 1 c − 1) 1 � 2( N 2 2 i ˆ tr Φ TL ( x ) a 4 g 2 a 2 Thus 3 � − 3 c − 1) 1 2( N 2 � � �L B i ( x ) � a 2 = 0 i =1 18
0.4 real part imaginary part 0.2 phase-quenched quenched 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 0 0.2 0.4 0.6 0.8 1 ag i =1 L B i ( x ) − 3 c − 1) 1 Expectation values of � 3 2 ( N 2 a 2 19
• The real part is consistent with the expected identity within 1 . 5 σ ( ⇒ strongly supports the correctness of our code/algorithm) • The imaginary part is consistent with zero • No notable difference of the phase-quenched average ( ⇒ systematic error due to wrong-sign determination is negli- gible) • Clear distinction from the quenched average ( ⇒ effect of dynamical fermions is properly included) • Effect of quenching starts at 2-loop ∼ g 2 ln( a/L ) 20
Another exact relation � � Q O 1 ( x ) � � = � �L B 1 ( x ) � � + � �L F 1 ( x ) � � = 0 21
0.25 real part imaginary part 0.2 phase-quenched quenched 0.15 0.1 0.05 0 -0.05 -0.1 0 0.2 0.4 0.6 0.8 1 ag Expectation values of L B 1 ( x ) + L F 1 ( x ) 22
• The relation is confirmed within 2 σ (note the difference in scale of vertical axis compared to the previous figure) • The quenched average is certainly inconsistent with the SUSY relation • No clear separation between the re-weighted average and the quenched one ( ⇐ The effect of quenching starts at 3- loop ∼ a 2 g 4 ln( a/L ) ) 23
Another relation 1 � � � �� � − iH ( x )ˆ � � Q O 2 ( x ) � � = + � �L F 5 ( x ) � � = 0 tr Φ TL ( x ) a 4 g 2 but H ( x ) = 1 2 i ˆ Φ TL ( x ) and thus 2 � �L B 2 ( x ) � � + � �L F 5 ( x ) � � = 0 24
0.4 real part imaginary part 0.2 phase-quenched quenched 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 0 0.2 0.4 0.6 0.8 1 ag Expectation values of 2 L B 2 ( x ) + L F 5 ( x ) 25
The situation is again similar with the last piece of the relation � � Q O 3 ( x ) � � = � �L B 3 ( x ) � � + � �L F 3 ( x ) � � + � �L F 4 ( x ) � � + � �L F 6 ( x ) � � = 0 26
0.4 real part imaginary part 0.2 phase-quenched quenched 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 0 0.2 0.4 0.6 0.8 1 ag Expectation values of L B 3 ( x ) + L F 3 ( x ) + L F 4 ( x ) + L F 6 ( x ) 27
So far, we have observed WT identities implied by the exact Q -symmetry of the lattice action The continuum theory is invariant also under other fermionic transformations, Q 01 , Q 0 and Q 1 Q 01 A µ = − ǫ µν ψ µ Q 01 ψ µ = iǫ µν D ν φ Q 01 φ = 0 Q 01 H = 1 Q 01 η = 2 H 2[ φ, η ] Q 01 χ = − 1 Q 01 φ = − 2 χ 2[ φ, φ ] 28
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