alexander ferling - uni- · pdf filesupersymmetric insights alexander ferling itp mun¨...
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supersymmetric insights
Alexander Ferling
ITP Munster
June 2, 2008
Alexander Ferling (ITP Munster) supersymmetric insights June 2, 2008 1 / 38
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1 Supersymmetric Hotspotsthe raise of supersymmetrythe particle spectrumthe consequencesconstructing a sym-theory
2 PHMC-Algorithmthe path-integralpolynomial approximationevaluating the trajectory
3 Action Improvementsspeed improvementssignal improvements
4 Matrix Inversionsthe reason whyconjugate gradient algorithmkrylov spacesmatrix deflationdomain decomposition
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Supersymmetric Hotspots the raise of supersymmetry
why supersymmetry?
aim of investigations: constructing a theory with only onesymmetry-group and delivering the standard model by symmetrybreaking
generators of the Poincare and Lie-Group I:
[Pµ, P ν ] = 0
[Mµν , P ρ] = i (ηνρPµ − ηµρP ν) ↔ [Ta, Tb] = fabcTc
[Mµν ,Mρσ] = i (ηνρMµσ + ηµσMνρ
−ηµρMνσ − ηνσMµσ)
→ non-trivial it is impossible(no-go theorem from Coleman & Mandula ’67)
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Supersymmetric Hotspots the raise of supersymmetry
to crack this theorem
extend the algebra by anti-commutating vectors instead ofcommutating ones Golfand & Likhtman (’71)
[even, even] = even
odd, odd = even
[even, odd] = odd
this so called Z2 graded algebra is the only one, consistent withrelQFT
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Supersymmetric Hotspots the raise of supersymmetry
the majorana-spinors Qα
anti-commuting values are associated with fermionic degrees offreedom → generators are majorana/weyl-spinors.
self-conjugated complex/real dirac spinors
ψCM = ψM
with commutation- and anticommutation relationsQα, Qβ
= 2σµ
αβPµ
Qα, Qβ =Qα, Qβ
= [Qα, Pµ] =
[Qα, Pµ
]= 0
[Qα,Mµν ] = σβµναQβ
[Qα,Mµν
]= σα
µνβQβ
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Supersymmetric Hotspots the particle spectrum
the particle spectrum
defining the Pauli-Lubanski-Vector Wµ
Wµ =1
2εµνρσP
νMρσ, Xµ = QσµQ
Y := Wµ − 1
4Xµ, [Yµ, Yν ] = imεµνσY σ,
(Y
m
)= y (y + 1)
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Supersymmetric Hotspots the particle spectrum
the particle spectrum
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Supersymmetric Hotspots the consequences
the consequences
solution for the hierarchy-problem
fewer divergencies
local supersymmetry φ′i (x) = U ji φj (x) → U j
i (x)φj (x)→ SUGRA (with ART in the low energy limit)
TOE’s only consistent with space-time supersymmetry
quark confinement
breaking electroweak interaction is a consequenceof supersymmetry breaking
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Supersymmetric Hotspots constructing a sym-theory
Wess-Zumino-Model ’74
take the action-functional
S [φ] =
∫d4xL (φ, ∂φ)
calculate the invariance under variation of the Poincare-Group P
δS [φ] =
∫d4xδL
with the extension (x1, x2, x3, x4, θ
1, θ2, θ1, θ2)
and transformations
θ → θ + η, θ → θ + η xµ → xµ + aµ − iησµθ + iθσµη
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Supersymmetric Hotspots constructing a sym-theory
the continuums lagrangian
equation of motion should be invariant under supersymmetrictransformations
∂
∂xµ
(∂L
∂ (∂φ/∂xµ)
)− ∂L∂φ
= 0
the simplest supersymmetric Lagrangian for the chiral multiplet
Lchiral = Lscalar + Lfermion = −∂µφ∗∂µφ− iψtσµ∂µψ
the interaction between fermion- and boson-fields should beyukawa-like and renormalisable. the product φ†φ leads to avektor-superfield
Super-Yang-Mills with symmetry breaking term
L = LSY M +mλλ
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Supersymmetric Hotspots constructing a sym-theory
on-shell action/Curci-Veneziano lattice action (’87)
...further constructions and restrictions...
leads to the euclidean on-shell continuum-action
SSY M =
∫d4x
1
4F a
µνFaµν +
1
2λaγµ∇µλ
a
putting it on the lattice leads to Slat = Sg + Sf with
Sg [U ] = β∑
x
∑µν
[1− 1
NcReTrUµν
]Sf
[U, λ, λ
]=
1
2
∑x
λ (x)λ (x) +
κ
2
∑x
∑µ
[λ (x+ µ)Vµ (x) (r + γµ)λ (x)
+λ (x)V Tµ (x) (r − γµ)λ (x+ µ)
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Supersymmetric Hotspots constructing a sym-theory
some comments on Q and V -matrix
by defining the Q-Matrix
Qy,x [U ] ≡ δyx − κ∑
µ
[δy,x+µ (1 + γµ)Vµ (x) + δy+µ (1− γµ)V T
µ (y)]
we can write Sf more compactly as
Sf =1
2
∑xy
λ (x)Qx,yλ (y) =1
2
∑xy
λT (x) CQx,yλ (y)
the adoint matrices have the form
[Vµ (x)]ab ≡ 2Tr[U †µ (x)T aUµ (x)T b
]
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PHMC-Algorithm
1 Supersymmetric Hotspotsthe raise of supersymmetrythe particle spectrumthe consequencesconstructing a sym-theory
2 PHMC-Algorithmthe path-integralpolynomial approximationevaluating the trajectory
3 Action Improvementsspeed improvementssignal improvements
4 Matrix Inversionsthe reason whyconjugate gradient algorithmkrylov spacesmatrix deflationdomain decomposition
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PHMC-Algorithm the path-integral
multi-bosonic representation
it’s not feasible to simulate Grassmann fields directly, becausee−SF = e−φDφ is not positive → poor importance sampling
we therefore integrate out the fermion fields to obtain the fermiondeterminant∫
[dλ] e−Sf =
∫[dλ] e−
12λQλ = ±
√detQ
questions concerning the ±-sign → ask Jaır
now we turn around - that thing is calles ”bosonification”, with√
detQ =[det(Q†Q
)] 14 . In QCD you have
det(Q†Q
)=
∫ [dφ†dφ
]exp
(−∑xy
φ†y
[Q†Q
]−1
yxφx
)
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PHMC-Algorithm polynomial approximation
polynomial approximation
use the approximation
limn→∞
Pn (x) =
[1
x
] 14
∀x ∈ [ε, λ]
keep in mind the condition number√λ/ε
choose the polynom
P(Q2)
= c0(Q− ρ1
) (Q− ρ2
). . .(Q− ρn
) (Q− ρ∗n
). . .(Q− ρ∗1
)so in our simulation, we have√
detQ =[det(Q†Q
)] 14 =
∫ [dφ†dφ
]exp
(−∑xy
φ†y(P(Q2))
yxφx
)
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PHMC-Algorithm evaluating the trajectory
the hamiltonian
update the field globally
takes large steps through configuration space
we introduce a fictitious Hamiltonian
H [P,U, φ] =1
2
∑xµj
P 2xµj + Sg [U ] + Sf [U, φ]
the action plays the role of a fictitious potential
HMC-Markov-Chain alternates two Markov-Steps:Molecular Dynamics Monte Carlo andMoment Refreshment (together they are ergodic)
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PHMC-Algorithm evaluating the trajectory
equations of motion
these are the hamilton equations of motion
dPxµj
dτ= −DxµjS,
dUxµ
dτ= iPxµjUxµ
to update of momenta and fields
U ′xµ = exp
∑j
i2TjPxµj∆τ
Uxµ, P ′xµj = Pxµj −DxµjS [U, φ] ∆τ
you have to derive the fermionic derivative ([DxµjVµ]ab = 2fbjc [Vµ]ac)
DxµjSf [U, φ] =n−1∑k=0
(φ
(k)1,a (x)
(DxµjQ
)φ
(k)†2,b (y)
)+
n−1∑k=0
(φ
(k)2,a (x)
(DxµjQ
)φ
(k)†1,b (y)
)
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PHMC-Algorithm evaluating the trajectory
integrators
Leapfrog integrator
Ttot (∆τ) = TP
(∆τ
2
)TU (∆τ)TP
(∆τ
2
)Sexton-Weingarten integrator
Tges (∆τ) = TU
(∆τ
6
)TP
(∆τ
2
)TU
(2∆τ
3
)TP
(∆τ
2
)TU
(∆τ
6
)higher order Leapfrog integrator with multiple timescales
Ti (∆τi) = TSi
(∆τi2
)Ti−1 (∆τi−1)Ni TSi
(∆τi2
)higher order Sexton-Weingarten integrator with multiple timescales
Ti (∆τi) = TSi
„∆τi
6
« Ti−1
„∆τi−1
2
«ffNi−1TSi
„2∆τi
3
« Ti−1
„∆τi−1
2
«ffNi−1TSi
„∆τi
6
«
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Action Improvements
1 Supersymmetric Hotspotsthe raise of supersymmetrythe particle spectrumthe consequencesconstructing a sym-theory
2 PHMC-Algorithmthe path-integralpolynomial approximationevaluating the trajectory
3 Action Improvementsspeed improvementssignal improvements
4 Matrix Inversionsthe reason whyconjugate gradient algorithmkrylov spacesmatrix deflationdomain decomposition
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Action Improvements speed improvements
speed improvements
two-step polynomial 1x ≡ Pn1,n2 (x) = P ′n1
(x)P ′′n2(x) with noisy
correction
eη†P ′′n2
(Q)η∫D [η] eη†P ′′
n2(Q)η
even-odd preconditioning (Q = Qγ5)
Q =
(γ5 −γ5κMeven−odd
−γ5κModd−even γ5
)→ det Q = det
(1− κ2MoeMeo
)determinant breakup det Q2 =
(det Q2
) 1nB
nB
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Action Improvements signal improvements
gauge action improvement
both terms can be optimized
S = Sg + Sf
↓ ↓DBW2 STOUT
a possible gauge action is
S = β11
∑plaq
ReTr
(1− 1
3Uplaq
)+ β12
∑plaq
ReTr
(1− 1
3Urect
)
Uplaq = q qq q
-?
6
Urect = q qq q
-?
-
6
Wilson TlSym Iwasaki DBW2β12 = 0 β12 = −1/12 β12 = −0.091 β12 = −1.4088
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Action Improvements signal improvements
STOUT link smearing
the (n+ 1)th stout smeared ”thick” link obtained iteratively from thenth level
U (n+1)µ (x) = eiQ
(n)µ U (n)
µ (x)
with
Qµ (x) =i
2
(Ω†µ (x)− Ωµ (x)
)− i
2NTr(Ω†µ (x)− Ωµ (x)
)and
Ωµ (x) = Cµ (x)U †µ (x)
the staples Cµ are defined as
Cµ (x) =∑ν 6=µ
ρµν
(Uν (x)Uµ (x+ ν)U †ν (x+ µ)
+U †ν (x− ν)Uµ (x− ν)Uµ (x− ν)Uν (x− ν + µ))
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Action Improvements signal improvements
some data from the analysis
0.573
0.574
0.575
0.576
0.577
0.578
0.579
0.58
0.581
0 100 200 300 400 500
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
STOUT ρ = 0.15, TlSym, Run 16c32, Nf = 1, β = 4.00 κ = 0.1450
<P>=0.57881(11), τint=7.3 thin plaquettethin eigenvalue
thick eigenvalue
0.535
0.54
0.545
0.55
0.555
0 500 1000 1500 2000 2500 3000 3500 40001e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
TlSym, Run 12c24, Nf = 1, β = 3.80, κ = 0.1710
Plaq./Minimal EV <P>=0.547840(67) τint=7.6
0.612
0.614
0.616
0.618
0.62
0 100 200 300 400 500 600
0.0001
0.001
0.01
0.1
1
10
100
1000
SYM on 24c48, TlSym β = 1.60 κ = 0.1500
<P>=0.617606(34), τint=2.3
thin plaquettethin eigenvalue
thick eigenvalue
0.62
0.622
0.624
0.626
0.628
0.63
0 500 1000 1500 20001e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
100
1000
SYM on 24c48, TlSym β = 1.60 κ = 0.1570
<P>=0.625979(31), τint=8.5
thin plaquettethin eigenvalue
thick eigenvalue
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Matrix Inversions
1 Supersymmetric Hotspotsthe raise of supersymmetrythe particle spectrumthe consequencesconstructing a sym-theory
2 PHMC-Algorithmthe path-integralpolynomial approximationevaluating the trajectory
3 Action Improvementsspeed improvementssignal improvements
4 Matrix Inversionsthe reason whyconjugate gradient algorithmkrylov spacesmatrix deflationdomain decomposition
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Matrix Inversions the reason why
the purpose of matrix inversions
examine the Gluino-Propagator
⟨Tλ (x) λ (x)
⟩= 〈T λ (x)λ (x)〉 C = 2
[δ2 lnZ [J ]
δJ (x) δJ (y)
]C
with the given partition function Z
Z =
∫D [λ] e−
12λCQλ.
we have to solve ⟨Tλ (x) λ (x)
⟩=
⟨Q−1 [U ]
⟩→ inversion of sparse matrices
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Matrix Inversions conjugate gradient algorithm
conjugate gradient algorithm I
instead of solving z = Q−1ω we solve
Qz = ω
(ω = a given source, Q = the fermion matrix, z = solution vector)
to find z, we use the conjugate gradient algorithm.
the basic idea of CG is, that equivalent to solve Qz = ω is extremising
E (z) := 〈ω, z〉 − 1/2 〈Qz, z〉.
the gradient of E at zk is
gk = ω −Qzk
conjugate gradient means now, to minimize E in a direction pk
instead of gk. This direction is Q-conjugated, which means
〈Qpi, pj〉 = 0
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Matrix Inversions conjugate gradient algorithm
conjugate gradient algorithm II
the algorithm step by step
1 take a source ω and set initially ω = z02 calculate the residuum
p0 = r0 = ω −Qz0
3 for n = 1, 2, . . .
an =|rn|2
〈pn, Qpn〉, zn+1 = zn + anpn, rn+1 = rn − anQpn
4 if |rn+1|2 < δ then the solution is zn+1, else calculate
bn = |rn+1|2 / |rn|2 , pn+1 = rn+1 + bnpn
and proceed with iteration
is valid only for positiv definite hermitian matrices, → extend to B†Ait can be used for any hermitian matrix A
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Matrix Inversions krylov spaces
krylov spaces I
consider a system of linear equations Ax = b and the residual vectorr ≡ b−Axi for an approximate solution xi
rewriting the system as
(I − (I −A))x = b
leads to basic iteration
xi = b+ (I −A)xi−1
= xi−1 + ri−1
= xi−2 + ri−2 + ri−1
...
= x0 + r0 + r1 + . . .+ ri−1.
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Matrix Inversions krylov spaces
krylov spaces II
multiply xi = xi−1 + ri−1 with A from the left
Axi = Axi−1 +Ari−1
and substract from b
b−Axi = b−Axi−1 +Ari−1
ri = ri−1 −Ari−1
= (I −A) ri−1
so finally we get
xi = x0 + r0 + (I −A) r0 + . . .+ (I −A)i−1 r0
= x0 +[r0, Ar0, A
2r0, . . . , Ai−1r0
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Matrix Inversions krylov spaces
krylov spaces III
this linear space defines the krylov subspace
Km(A, r) = spanr,Ar, . . . , Am−1r.
convergence is measured by the residual rn = |b−Axn|.more specificially, we seek an approximate solution xn in Kn byimposing the petrov-galerkin condition
rn ≡ b−Axn⊥Ln
where Ln is an n-dimensional subspace
two broad choices:
Ln = Kn (A; r0) ↔ orthognoalisation (Arnoldi, GMRES, CG, GCR...)Ln = Kn
(A†; r0
)↔ bi-orthognoalisation (Lanczos, BCG, BiCGstab...)
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Matrix Inversions matrix deflation
on which circumstances is matrix deflation feasible?
stochastic estimator technique⟨η†iZi
⟩Nest
Nest→∞= Q−1
ii
collect informations about Q in each CG forthe next step
feed CG with a
galerkin-projected vector
x0 = W(W TAW
)−1W T b.
→ convergence will raise
SET
Qz1 = η1α
↓Qz2 = η2α
↓Qzi = ηiα
...QzN = ηNα
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Matrix Inversions matrix deflation
deflation: the stathopoulos-orginos algorithm
InitCG
Algorithm 1: Basisiteration
iterative solution of Ay = c
Initialisation
choose y0;
s0 = c − Ay0;
ω0 = s0;
Iteration
for j = 0, 1, ... until covergence do
γj =`sj , sj
´/
`ωj , Aωj
´;
yj+1 = yj + γjωj ;
sj+1 = sj − γjAωj ;
δj+1 =`sj+1, sj+1
´/
`sj , sj
´;
ωj+1 = sj+1 + δj+1ωj ;
end do
Algorithm 2: InitCG
iterative solution of Ax = b
Initialisation
choose x−1;
r−1 = b − Ax−1;
x0 = x−1 + W“
WT
AW”−1
WT
r−1;
r0 = b − Ax0;
p0 = r0;
Iteration
for j = 0, 1, ... until covergence do
αk = (rk, rk) / (pk, Apk) ;
xk+1 = xk + αkpk ;
rk+1 = rk − αkApk ;
βk+1 =`rk+1, rk+1
´/ (rk, rk) ;
pk+1 = rk+1 + βk+1pk ;
end do
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Matrix Inversions matrix deflation
eigCG
generate an initial V withrestarting-CG
Tm =(W TAW
)−1is the
lanczos-matrix
in 8. and 9. we useraileigh ritz, to computean orthonormal ritz basisfor space [Y, Y ]
return nev ritz vectorsfrom V
1. V = [ ] ;
2. for j = 0, 1, ... until covergence do
3. standard CG iteration
4. update three elements of Tj
5. if (size (V, 2) == m)
6. solve TmY = Y M, for nev lowest eigenpairs
7. solve Tm−1Y = Y M, for nev lowest eigenpairs
8. [Q, R] = qr`ˆ
Y, Y , 0˜´
, and H = QH
TmQ
9. solve HZ = ZM for 2 nev lowest eigenpairs
10. Restart: V = V (QZ) and T2nev = M
11. set the 2nev + 1 column of T2nev+1 as VH
Arj
12. endif
13. V =ˆV, rj/‖rj‖
˜14. end CG
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Matrix Inversions matrix deflation
solver convergence
Figure: convergence of the solvers. the blue ones are the 24 incremental eigCGiterations, red the last 24 init-CG iterations
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Matrix Inversions domain decomposition
deflation: the luscher algorithm
at the beginning of each MD-trajectory, there will be fermion-fieldsφl (x) , l = 1, . . . , Ns stochastically gernerated through a so calledsmoothing procedure
then, they will be projected on non-overlapping Blocks Λ with
φΛl (x) =
φl (x) wenn x ∈ Λ,
0 sonst
Figure: often used blocksize 44
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Matrix Inversions domain decomposition
mode projection I
a given field ψ can be projected with an orthogonal projector P on thespace S, which is spanned by the orthonormalbasis φ1 (x) , . . . , φN (x)
Pψ (x) =N∑
k=1
φk (x) (φk, ψ) .
the complete system is combined by the ”inner” system S and an”outer” complementary System S⊥
ψ (x) = χ (x) +N∑
k,l=1
φk (x)(A−1
)kl
(φl, η)
here we used the so called little Dirac-Operator
Akl = (φk, Dφl) , k, l = 1, . . . , N.
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Matrix Inversions domain decomposition
mode projection II
Figure: deviation of the smallesteigenvalues
Figure: approximation of thelowest modes by a constant
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Matrix Inversions domain decomposition
comparison of the various methods
Morgan/Wilcox Stath./Orginos Luscher
Solver GMRES / BiCGStab CG GCR
Matrix Type non-herm. herm. non-herm.(Algebraic) (Algebraic) (Lattice)
Simultaneous yes yes nosolveEigenvalue every cycle (GMRES) every cycle, everyuse for beginning (BiCGStab) beginning (s ≤ s1) outermultiple rhs’s restart (s > s1) iteration
Algorithm mild large smallacceleration
Table: some points of comparison for the three algorithms considered
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Matrix Inversions domain decomposition
Kapitel1: Supersymmetric Hotspots
D. Talkenberger, Monte-Carlo-Simulationen von Modellen derElementarteilchenphysik mit dynamischen Fermionen, Dissertation,Universitat Munster, Oktober 1997.
K. Spanderen, Monte-Carlo-Simulationen einer SU(2)Yang-Mills-Theorie mit dynamischen Gluinos, Dissertation, UniversitatMunster, Oktober 1998.
C. Gebert, Storungstheoretische Untersuchungen der N = 1supersymmetrischen Yang-Mills-Theorie auf dem Gitter, Dissertation,Universitat Munster, 1999
S. Luckmann, Supersymmetrische Feldtheorien auf dem Gitter,Dissertation, Universitat Munster, 2001
K. Muller, Einfuhrung Supersymmetrie, Primer, Universitat Zurich,Oktober 2002.
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Matrix Inversions domain decomposition
Kapitel2: PHMC-Algorithm
I.Campos et.al. , Monte Carlo simulation of SU(2) Yang-Mills theorywith light gluinos, [arXiv:hep-lat/9903014]
R. Peetz, Spectrum of N = 1 Super Yang Mills Theory on the Latticewith a light gluino, PhD, 2003
M. Luscher, A new approach to the problem of dynamical quarks,Nuc.Phys. B418, S.637-648, 1994
I. Montvay, E.E. Scholz, Updating algorithms with multi-stepstochastic correction, [arXiv:hep-lat/0506006]
B. Gehrmann, The step scaling function of QCD at negative flavornumber, PhD, 2002
E.E. Scholz, Light Quark Fields in QCD: Numerical Simulations andChiral Perturbation Theory, PhD, 2005
E.E. Scholz, I. Montvay, Multi Step stochastic correction in dynamicalfermion updating algorithms, [arXiv:hep-lat/0609042v2]
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Matrix Inversions domain decomposition
Kapitel3: Action Improvements
Ph. Forcrand et.al., Search for Effective Lattice Action of Pure QCD,[arXiv:hep-lat/9608094v1]
K. Jansen, I. Montvay, C. Urbach, et.al., Stout Smearing for TwistedMass Fermions, [arXiv:hep-lat/90709.4434v1]
C. Morningstar, M.J. MPeardon, Analytic smearing of SU(3) linkvariables in lattice QCD, [arXiv:hep-lat/0311018]
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Matrix Inversions domain decomposition
Kapitel4: Matrix Inversions
Y. Saad, Iterative Methods for sparse linear systems, SIAMPhiladelphia, PA, USA, 2003
K. Spanderen, Monte-Carlo-Simulationen einer SU(2)Yang-Mills-Theorie mit dynamischen Gluinos, Dissertation, UniversitatMunster, August 1998
G. Munster, I. Montvay, Quantum Fields on a Lattice, CambridgeMonographs on Mathematical Physics
A. Stathopoulos, K. Orginos, hep-lat/0707.131v1
M. Luscher, Local coherence and deflation of the low quark modes inlattice QCD hep-lat/0706.2298v4
R.B. Morgan, W. Wilcox, Deflated Iterative Methods for LinearEquations with Multiple Right-Hand Sides math-ph/0405053v2
U. Wenger, Introduction to Lattice QCD algorithms, Lattice PracticeTalk, 27.11.06, Zeuthen
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