The flavour projection of staggered fermions and the quarter-root trick
aa r X i v : . [ h e p - l a t ] J un JHEP06(2007)048
Published by Institute of Physics Publishing for SISSA/ISAS
Received:
Accepted:
The flavour projection of staggered fermions and thequarter-root trick
Steven Watterson
Department of Mathematics, Trinity College, Dublin 2, Ireland andDivision of Pathway Medicine, University of Edinburgh Medical School, Chancellor’sBuilding, 49 Little France Crescent, Edinburgh, EH16 4SB, ScotlandE-mail: [email protected]
Abstract:
It is shown that the flavour projection of staggered fermions can be writtenas a projection between the fields on four separate, but parallel, lattices, where the fieldson each are modified forms of the standard staggered fermion field. Because the staggeredDirac operator acts equally on each lattice, it respects this flavour projection. We show thatthe system can be gauged in the usual fashion and that this does not interfere with flavourprojection. We also consider the path integral, showing that, prior to flavour projection, itevaluates to the same form on each lattice and that this form is equal to that used in thequarter-root trick. The flavour projection leaves a path integral for a single flavour of fieldon each lattice.
Keywords:
Lattice Quantum Field Theory, Lattice Gauge Field Theories, Lattice QCD. c (cid:13) SISSA/ISAS 2019 http://jhep.sissa.it/archive/papers/jhep062007048/jhep062007048.pdf
HEP06(2007)048
Contents
1. Introduction 12. Background 23. Flavour projection 34. Chiral symmetry 45. Gauging the links 46. Path integral formulation 57. Conclusion 7
1. Introduction
Constructing lattice field theories that incorporate a single non-degenerate fermion field,with chiral symmetry, has proven to be very challenging. Wilson originally identifiedthe problem by showing that na¨ıvely discretizing the continuum Dirac equation led to adoubling in the number of fermion fields in each dimension of momentum space [1]. Asa solution, he proposed adding a term proportional to the second derivative to break thedegeneracy between the physical field and the doublers. However, this term broke the chiralsymmetry of the theory. Subsequently, Kogut, Susskind and Banks proposed the staggeredfermion formulation [2][3][4] in which the number of degenerate fields is reduced to 2 n/ in n dimensions and which retains a limited form of chiral symmetry. Other formulationshave since succeeded in isolating a single chiral field by introducing an infinite number ofregulating fields [5][6][7]. In the finite case, these formulations contain an exponentiallysuppressed term that breaks the chiral symmetry of the theory.For reasons of computational efficiency, the staggered fermion formulation has re-mained popular in lattice calculations and it is common to see the quarter-root trick usedto model a system with one flavour of fermion [8][9]. It has been shown that, in the free fieldcase, it is possible to construct a lattice Dirac operator whose determinant is the quarter-root of the determinant of the conventional staggered Dirac operator [10]. Although, thelocality of this operator has been questioned [11], numerical studies suggest that a suitablylocal operator does exist [12].In this paper, we draw on previous work [13] to show that the flavour projection ofstaggered fermions can be written as a projection between four modified staggered fermionfields on four separate, but parallel, lattices. The staggered Dirac operator respects the– 1 – HEP06(2007)048 flavour projection because it acts upon the fields on each lattice equally. We show thatwe can gauge the fields in each lattice in the usual way and that the flavour projectionis equally valid in the gauged case. We discuss the role of chiral symmetry and we seethat, in this framework, the γ ⊗ γ chiral symmetry of the staggered fermion formulationcan be used to isolate the chiral components of the fields. Finally, we consider the pathintegrals on each lattice, before and after flavour projection, showing that projection leavesthe path integral for a single flavour of field on each lattice. In considering this, we see thatevaluating the path integral, prior to flavour projection, leads to an expression identical tothat used in the quarter-root trick.
2. Background
Our starting point is the staggered fermion formulation in four Euclidean dimensions. Thelattice Dirac equation is diagonalised by rewriting the spinor field as ˜ ψ n , where˜ ψ n = T ( n ) † ψ n and T ( n ) = γ n γ n γ n γ n . We retain only the first component of ˜ ψ n , defining the field χ n = ˜ ψ n . The resultingmassless action is S = 12 X n,µ η µ ( n ) ¯ χ n [ χ n +ˆ µ − χ n − ˆ µ ] , where η ( n ) = 1 η µ ( n ) = ( − n + .. + n µ − : for µ = 1 . The lattice sites are grouped into hypercubes of side twice the original lattice spacing,each containing sixteen sites. We relabel the coordinate of the lattice site n with thecoordinate of the origin of the hypercube, m , and the location of the site within thehypercube, s . The relationship between the four component vectors n , m and s is n = m + s .Together the hypercubes yield a fermion field that is free from degeneracy in momentumspace ˆ ψ m = N X s T ( s ) χ sm , (2.1)where N is a normalisation constant. For N = √ , the action is S = X m,µ b ¯ˆ ψ m (cid:20) ( γ µ ⊗ I ) ∂ µ + b γ ⊗ γ ∗ µ γ ) (cid:3) µ (cid:21) ˆ ψ m , (2.2)where A ⊗ B is the product of the spin and flavour spaces and b is twice the original latticespacing.ˆ ψ m is now a four by four matrix containing four degenerate flavours. The action mixesboth the spin and flavour space and the only operator that anticommutes with both termsin the action is γ ⊗ γ , generating a limited form of chiral symmetry.– 2 – HEP06(2007)048
3. Flavour projection
In the continuum limit, the second term in equation (2.2) goes to zero, we have thatˆ ψ m → ˆ ψ ( m ) and the columns of ˆ ψ ( m ) come to represent separate flavours of the field.Each column can be isolated with the operator ˆP ( b ) , where ˆP ( b ) ˆ ψ ( m ) = ˆ ψ ( m ) P ( b ) and the projection matrix P ( b ) is of the formdiag(1 , , ,
0) : for b = 1 , diag(0 , , ,
0) : for b = 2 , diag(0 , , ,
0) : for b = 3 , diag(0 , , ,
1) : for b = 4 .P ( b ) is defined through P ( b ) = 14 (1 + iα b γ γ ) (1 + β b γ γ γ γ ) (3.1)with α b = ( − , +1 , − , +1) T β b = ( − , − , +1 , +1) T . We can expand the projection matrix to give P ( b ) = 14 (1 + iα b γ γ + β b γ γ γ γ − iα b β b γ γ ) (3.2)and this is equal to P ( b ) = (cid:16) diag(1 , , ,
1) + α b diag( − , , − , β b diag( − , − , ,
1) + α b β b diag(1 , − , − , (cid:17) . Writing P = diag(1 , , , , P = diag( − , , − , ,P = diag( − , − , , , P = diag(1 , − , − , , we can see that [ D, ˆP ( b ) ] = 0, where D is the staggered Dirac operator, because[ Dψ ( m )] P i = D [ ψ ( m ) P i ] . At finite lattice spacing, b = 0 and the second term in the staggered Dirac operatorbreaks this associativity. Thus [ D, ˆP ] = 0. However, if we could transform the P i in amanner consistent with the effect of D on ˆ ψ m , we would be able to maintain [ D, ˆP ( b ) ] = 0at finite lattice spacing.To implement this strategy, we write ˆ ψ m P ( b ) asˆ ψ m P ( b ) = ˆ ψ m ( P + α b P + β b P + α b β b P )= (cid:16) ˆΨ m + α b ˆΨ m + β b ˆΨ m + α b β b ˆΨ m (cid:17) . We introduce three additional lattices, in parallel to the first, and on the i -th lattice, L i ,we place the field ˆΨ im . Flavour projection now becomes a projection between the lattices– 3 – HEP06(2007)048 and to describe it we introduce the following operator to map a field from lattice L i tolattice L j ˆ T ij : L i → L j . Writing the fermion field across all four lattices as ˆΨ m = P i ˆΨ im and using equation (3.1),the flavour projection operator can now be written as ˆP ( b ) ˆΨ m = (cid:16) β b ( T − T + T − T ) (cid:17) × (cid:16) α b ( T − T + T − T ) (cid:17) ˆΨ m . Application of ˆP (1) ˆΨ will leave column i on L i . However, because D mixes the spinand flavour spaces equally on all four lattice, the same degrees of freedom of ˆ ψ m will beprojected out, irrespective of the order in which we apply D and ˆP ( b ) to ˆ ψ m . Thus we have[ ˆP ( i ) , D ] = 0.
4. Chiral symmetry
With the components from only one flavour of field on each lattice, we can see that theresidual γ ⊗ γ chiral generator is sufficient to isolate the chiral components. For example,the application P R ˆP (1) ˆΨ leaves the positive chirality components of the first column ofˆ ψ m on L and of the second column of ˆ ψ m on L . It also leaves the negative chiralitycomponents of the third column of ˆ ψ m on L and of the fourth column of ˆ ψ m on L .Because the chiral projection isolates the same components of ˆΨ im on each lattice, it doesnot interfere with the flavour projection and so we have that [ P R/L , ˆP ( b ) ] = 0.The effect of the projection on the chiral anomaly can be understood by considering ˆP (1) P R ˆΨ. The application of P R to ˆΨ m projects out two positive and two negative chiralityspinors on each lattice which ensure that the chiral current is conserved. Writing ˆP ( b ) as ˆP ( b ) = ˆP ( b ) β ˆP ( b ) α ˆP ( b ) β = (cid:16) β b ( T − T + T − T ) (cid:17) ˆP ( b ) α = (cid:16) α b ( T − T + T − T ) (cid:17) , we can see that ˆP (1) α removes one positive and one negative chirality spinor from each lattice,leaving one positive and one negative chirality spinor behind. At this stage the chiralcurrent is still conserved on each lattice. The subsequent application of ˆP (1) β , separatesthe remaining spinors, placing them on different lattices. This results in a chiral currentflowing between L and L and between L and L which manifests itself as an anomalyon each lattice.
5. Gauging the links
The gauging of staggered fermion fields takes place in the χ n basis. However, the modifiedform of ˆ ψ m that we place on each lattice leads us to introduce modified forms of χ sm .– 4 – HEP06(2007)048
Using the identity
T r ( T ( n ) † T ( m )) = 4 δ n,m , the inverse of equation (2.1) can be shownto be χ sm = 14 N T r (cid:16) T ( s ) † ˆ ψ m (cid:17) . We define the modified field for lattice i as ω ism = 14 N T r (cid:16) T ( s ) † ˆΨ im (cid:17) . (5.1)By writing the diagonal matrices in terms of the defining γ -matrix combinations fromequation (3.2), we can relate ω ism to χ sm as follows. ω sm = N T r (cid:16) T ( s ) † ˆ ψ m (cid:17) = χ sm ω sm = i N T r (cid:16) T ( s ) † ˆ ψ m γ γ (cid:17) = − iρ ( s, χ C sm ω sm = N T r (cid:16) T ( s ) † ˆ ψ m γ γ γ γ (cid:17) = ρ ( s, χ C sm ω sm = − i N T r (cid:16) T ( s ) † ˆ ψ m γ γ (cid:17) = iρ ( s, χ C sm . (5.2)Here ρ ( M,N ) is defined to be ( − ν , where ν is the number of pairs ( m, n ) with m ∈ M , n ∈ N and m > n [14]. We have also introduced C as a complementarity operator. C s gives a vector that complements s in the { , } subspace, but is the same as s in the { , } subspace. Similarly, C s gives a vector that complements s in the { , } subspace, butis the same in the { , } subspace. C s gives a vector that complements s in all fourdimensions.On each lattice ω ism is gauged in the same manner as χ sm . Because of the linearity ofthe transform in equation (5.1), we can apply the projection operator ˆP ( b ) to ω = P ims ω ism to project a single gauged spinor onto each lattice.
6. Path integral formulation
The path integral contains contributions from all four lattices. On each lattice, the pathintegral takes the form Z [ dA ] Z [ d ¯ˆΨ i ][ d ˆΨ i ] e − P xx ′ T r „ ¯ˆΨ ix ′ K x ′ x ( A ) ˆΨ ix « ! L , (6.1)where L is a constant yet to be determined. The fields on each lattice share the samedegrees of freedom, but, because each lattice lies in a separate space, we can separatelyevaluate the contribution from each.The number of elements of ˆΨ , ˆΨ and ˆΨ that are the negative of their counterpartsin ˆΨ is even and this gives the following equivalence[ d ¯ˆΨ i ][ d ˆΨ i ] = [ d ¯ˆΨ ][ d ˆΨ ] = [ d ¯ˆ ψ ][ d ˆ ψ ] , giving us the same fermionic measure of integration on each lattice.– 5 – HEP06(2007)048
If we consider the action on each lattice written as a matrix equation, it takes the form S i = ¯ˆΨ iα K αβ ( A ) ˆΨ iβ , where K αβ is the gauged staggered Dirac operator. By rewriting theaction as ¯ˆΨ iα K αβ ( A ) ˆΨ iβ = ¯ˆΨ α K iαβ ( A ) ˆΨ β = ¯ˆ ψ α K iαβ ( A ) ˆ ψ β , we introduce K iαβ ( A ) in which an even number of columns are equal to the negative oftheir counterpart in K αβ ( A ). Evaluating (cid:18)Z [ d ¯ˆ ψ ][ d ˆ ψ ] e − P αβ ¯ˆ ψ α K αβ ( A ) ˆ ψ β (cid:19) L gives (cid:0) det [ K ( A )] (cid:1) L which is equal to ( det [ K ( A )]) L . Because K αβ ( A ), K αβ ( A ) and K αβ ( A ) differ from K αβ ( A )by an even number of columns multiplied by −
1, we have that (cid:18)R [ d ¯ˆ ψ ][ d ˆ ψ ] e − P αβ ¯ˆ ψ α K iαβ ( A ) ˆ ψ β (cid:19) L = (cid:0) det [ K i ( A )] (cid:1) L = (cid:0) det [ K ( A )] (cid:1) L = ( det [ K ( A )]) L . (6.2)Hence, the products of all four contributions to the path integral is( det [ K ( A )]) L . For the path integral to maintain the correct continuum limit, we require that L = . Thisgives an expression on each lattice that is of the same form as that used in the quarter-roottrick (equation (6.2)).If we evaluate the path integral in the ω basis, we find from equation (5.2) that, foreach site m on each lattice i , the sixteen components of ω ism contain an even number ofterms that are the negative of their counterpart amongst χ sm . Hence, we have that[ d ¯ ω i ][ dω i ] = [ d ¯ ω ][ dω ] = [ d ¯ χ ][ dχ ] . As in the ˆ ψ basis, the − det [ M ( A )]) L on each lattice, where M ( A ) is the Dirac operator in this basis.If, instead of immediately evaluating the path integral on each lattice, we first apply ˆP ( b ) to equation (6.1), the flavour projection will affect both the action and fermionicmeasure of integration. As discussed, on each lattice, it will leave the action acting ona single flavour of field. It will also project between the measures of integration on eachlattice, to leave the components belonging to the same single flavour as in the action. Theresulting path integral on each lattice, describes a form of single gauged fermion.– 6 – HEP06(2007)048
7. Conclusion
In this paper, we have shown that it is possible to describe flavour projection in staggeredfermions as a projection between four lattices where modified forms of the fermion fieldoccupy each lattice. The staggered Dirac operator respects this flavour projection becauseit mixes the spin and flavour spaces of the fields on each lattice equally. We saw that, afterflavour projection, the γ ⊗ γ chirality of the staggered fermion formulation is sufficientto isolate the chiral components of the remaining fields. We also showed that this systemcould be gauged in the usual way and that it was possible to project out a single gaugedfield. We showed that the projection could be applied to the full path integral and that,prior to flavour projection, evaluating the path integral lead to a form consistent with thequarter-root trick commonly seen in lattice QCD.It is our hope that this work will contribute to the discussion on fermion doubling andthat, by finding a natural home for the quarter-root trick, this formulation can contributeto the current debate on its validity. References [1] K G Wilson,
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