JHEP10(2000)039 October 23, 2000 Accepted: uscher’s gauge-invariant lattice formula- October 6, 2000, Received: Renormalization Regularization and Renormalons, Lattice Gauge The Weyl fermion belonging to the real representation of the gauge group [email protected] HYPER VERSION Department of Mathematical Sciences, IbarakiMito University 310-8512, Japan E-mail: Abstract: provides a simple illustrativetion example for of L¨ chiral gaugemeasure globally theories. over the gauge-field Wesector; configuration can space there explicitly in the is construct arbitrary noshown the topological that global fermion this obstruction Weyl integration formulation correspondingMajorana is equivalent to (left-right-symmetric) to fermion, the a in lattice Wittengiven which formulation anomaly. by the based the fermion on It pfaffian the partitionThis with is function observation a is suggests definite a sign, naturaldifferent up relative topological normalization to sectors of physically for the irrelevant the fermiontation. contact Weyl measure terms. fermion in belonging to the complex represen- Keywords: Hiroshi Suzuki Field Theories, Gauge Symmetry, Anomalies in Field and String Theories. Real representation in chiralon gauge the theories lattice JHEP10(2000)039 real This is the 3 The local anomaly is 2 topological obstruction [1, 3, 9], gauge anomaly that corresponds to global local 1 uscher’s formulation 2 gauge invariance has emerged recently [1]–[6]. In this exact of the gauge group. Our motivation is two fold: 1 -real representations of SU(2), from which the local anomaly is also absent, it has pseudo In the formulation of [1, 3], there are two kinds of obstruction that prevent the This is sometimes calledThe the Witten real-positive anomaly representation in in the lattice literature. gauge theoryFor has been studied also from the viewpoint of the 1 2 3 expected result from the knowledge in the continuum theory [10, 12, 13]. We will absent from real representations, soare we highlighted. expect that Inrepresentations global and fact, issues that we it in is the can always possibleintegration formulation show to measure construct that the globally, gauge-invariant there over fermion the is gauge-field no configuration global space. obstruction for real 1. Introduction A general strategy towhile implement preserving anomaly-free the chiral gauge theories on the lattice representation 3. Global existence of the4. fermion integration measure Fermion expectation values5. Matching to the Majorana6. formulation Relative normalization for different7. topological sectors Conclusion 4 11 9 paper, we apply this formulation to a 6 single Weyl fermion, which belongs to a 12 Contents 1. Introduction2. Real representations in L¨ 1 spectral flow [11]. been shown [9] that a globally consistent definition of the fermion integration measure is impossible. gauge-invariant formulation. The first is the the gauge anomaly in thespacings continuum [1]–[7] theory, (see but also requires [8]). a The control second with is finite the lattice which is a lattice counterpart of the Witten anomaly [10]. JHEP10(2000)039 , ∗ 5 + γ D (2.1) (2.2) 5 = γ T 5 γ = , .  .Wetakethe 1 ) B C − in the fermion x − ( C 5 T O = ψ Cγ T 1 , − B ∗ µ B γ ) − x ( = ψ =1and . T µ im F γ B S † 2 1 − − B e − , = ∗ µ O ) γ 1 ] x − ( ψ = C 4 T µ µ ]D[ Bψ γ ) 2 ψ Cγ = x ( 1 D[ uscher’s formulation T − from the charge conjugation matrix B Z be gauge-covariant and that it depends, locally 5 µ = imψ representation. Cγ D Bγ 2 1 F = )+ B satisfies the Ginsparg-Wilson (GW) relation [19] hOi x ( D complex Dψ ) x . These imply ( C ψ  − is defined by = x . We require that B X T 4 D C 5 a = aDγ F S = =1and Secondly, there has been a renewed interest [14] in the context of the domain We begin with recapitulating some basics of the formulation. For unexplained The matrix 5 4 C † In this paper, the fermion action is taken as Dγ where the Dirac operator In the formulation of [1, 3], the expectation value of an operator 2. Real representations in L¨ explicitly construct such athe measure quantities in and, thetopological with sectors. formulation, that In including measure, this fermionfor we way, real the expectation can formulation. representations values work provide in an out illustrative general all example wall fermion [15] onwhich a the lattice fermion formulation (the ofrepresentation. gaugino) SUSY Usually belongs Yang-Mills such theories to a [16]–[18], the fermioneither in is real Weyl regarded representation, or as i.e. Majorana thetheory the Majorana is and adjoint fermion a the because latter matternot is of obvious more convention whether symmetric in or with not four-dimensionaland respect the continuum that lattice to based formulation the based on chirality. on theexpectation However the values Majorana it Weyl in fermion fermion is general [1]–[6] are topologicalby equivalent. sectors contact We differ, will terms in show that the thatphysically are two formulations, fermion irrelevant equivalent. only in physical This amplitudes.provides result a Thus supports unified they treatment the aretween of actually view the chiral Weyl that and gauge the the theoriesnormalization Majorana framework in of formulations general. of the moreover suggests fermion [1, The athe integration 3] natural matching measure Weyl relative be- fermion in in different the topological sectors for and smoothly, on the gaugelocality field. and the Such smoothness a are, Dirac however, guaranteed operator only in in fact a exists restricted [20, gauge-field 21]. The sector is defined by the path integral notations and for more details,throughout see this [1, paper. 3]. We assume that the lattice volume is finite representation of the Dirac algebra such that C JHEP10(2000)039 . j = = ≡ v B k ∗ ∗ T − c = (2.4) (2.3) ˆ D P D d j k v and the and the )isthe = a − Q j k ˆ c P c T B . For real ) ( ∗ − jk x R ˆ ]= R ( P Q j ψ v The GW chiral k = j , one has 6 P j − P e v ˆ P as D[ = , we can consistently B j ,where + ). e − c )= p, T x ˆ P x ( P ) ( a ϕ D ) ψ ] in (2.1), one first intro- T x ( ( ψ is any fixed positive number . These conditions, however, = † R . How to choose (and whether −  ψ , j ˆ P ]D[ − e x D c kj d ψ + Tr ) P j c 1 P 4 d − a Q . Note that the mass terms in (2.2) )= Q a ψ Q ··· )( T )= 2 ( The chiral projectors are then defined x = for all plaquettes c ( R ), which satisfy the constraint d k ψ,ϕ + 3 1 7 − v 2. Since and the charge conjugation property c =det ˆ / = P ). d ) k j 5 ψP 5 X c ∗ Tr ≡ γ ) aD d < a Dγ j j 5 ± c and T k − )= . γ Q ( d 5 x )] ,..., γ j ψ ( R p (1 j = 2 ( 5 Q e v =ˆ , = =(1 † γ U † [ ) D ± ψ 5 is the complex conjugate representation of R ]= = =1 γ P − . The coefficients are thus related as ψ ∗ ˆ P j − 5 ( R Q γ 1 j k v The fermion field is then expanded as 2and 8 / =1and(ˆ . ) 30. Under this admissibility, the gauge-field configuration space is 2 5 where jk ) / ˆ γ 0 limit at the very end of calculations. The mass terms are consistent δ -hermiticity 5 5 5 γ , ± γ 1 → − )= k is the representation of the gauge group and B m for real representations. ∗ ,v 1 =(1 R R j − v → ± To define the fermion integration measure D[ The chirality of the Weyl fermion is introduced as follows [26]: In (2.2), we have introduced the “Majorana” mass terms to treat topologically ˆ R The inner product for spinors is defined by ( Throughout this paper, the complex conjugation and theFor transpose definiteness, operation we on will consider anNote the operator that left-handed (ˆ Weyl fermion. P 8 5 6 7 with a unitary matrix are also consistent with this definition of the chirality because where smaller than 1 divided into topological sectorsassume [24, the 1] (see also [25]). As further requirements, we impose the chirality as by BD BDB it is possible to choose)sistent the with phase the over the gauge gauge-field invariance is configurationfor the space the central that issue is anti-fermion in con- is the defined formulation. similarly The measure but with respect to duces basis vectors measure is defined by D[ representations, we may take measures differ by a phase factor, matrixisdefinedbyˆ and ( configuration space, as expectedoperator from [21], the the index sufficient relation condition [22]. is For [23] the overlap-Dirac representation matrix for the Lie algebra of the gauge group. This implies do not specifydepend the on measure the uniquely; gauge field. there For remains a different a choice phase of basis ambiguity vectors that may with the Fermi statistics and, for real representations, gauge-invariant. non-trivial sectors, in which there areas zero the modes vacuum of the sector. Diractake operator, the If just one as easily is interested in the massless theory, it is sufficient to are defined with respect to the corresponding kernel in position space. JHEP10(2000)039 1 1 ). is = = v − 1 l )= ∗ 3 2 0 2 − 0 1 v ˆ v v (2.5) ,u P )=0 1 )and 1 in the 1 ,v − + u = 1 4 ( ,v v P B l and 3 u 2 is linearly q ,andthis v .Thiscan v 2 / = 4 Tr 2 − v 1 v ˆ v 0 2 ) P u and ( 3 v ) for different 0 1 )=( = ,u 4 v ,..., M and 2 1 ( 2 v ,v v 1 , ϑ = ( 2 v ]isTr and (say) 0 , 1 v such that ψ − v 3 M F =1 j − 1 u v ˆ , where we may assume v P j ) k 3 ( u hOi = inthecompletesetby j ) k )=0,( j ,u k 2 M 2 v 1 v ( u v − .Nowweseethat 6=1 ,v iϑ ( ˆ j ∗ ). We will construct the basis 3 .Wefirstset ) = 1). Note that 3 = P e satisfies j 2 v ) v − P 1 ( u 0 1 + and 3 ,v − v M x, µ − P 2 = u ( ( = 1 k B v with 2 u j v U N = v j u = )= G v 3 ), − 4 0 S 3 e ,wemayreplace v ˆ x v P − j 4 ,v ( e u 1 k ] 0 j = ) and the relative phase , because v v stands for each connected component in the k ∗ 1 U k 4 3 v c M v , 1 ( 2 k D[ M such that it is orthogonal to , − N by 3 P M 6=1 B ] can be chosen as being independent of the gauge j u Z 3 ψ v ≡ P M X 0 1 )= , because ( v = x 3 from 1 ( Z v =1and . Clearly this procedure can be repeated pairwise and we 4 = 3 4 ψ v i v v = 2 1 is constant within a connected component in the admissible v h is correctly normalized, ( and − 2 2 as ˆ v hOi and P v ( 2 3 , in the constrained space v 1 v ,where j B is defined from v + u span a complete set, we have 4 − P v j . The phase of D[ Tr = u kl c ); ) = 0. Since δ 3 introduced in the previous section starting with a complete set of arbitrarily T d 4 is chosen as e v j , B ,v v 3 Z ··· 3 . Next, we define e )= v v 2 2 † l ( c v Take a certain gauge-field configuration An important point to note is that the above construction refers to a specific v .Since 6= 0 without loss. Thus, we can replace q d , 1 1 † k / v c 2 3 v e where complete set by are left with the orthonormal complete set − number depends on theviolation gauge-field in configuration. topologically non-trivial In sectors is this naturally way, incorporated. the Since fermion-number Tr ( k and ( admissible space. Thespace may restriction be of implemented by theother the modified hand, gauge-field plaquette as integration action already [1]to to for emphasized fix the example. in the admissible On [1], the relative at normalization the moment there is no obvious way topological sectors. We will come back to this point in a later section. 0, since chosen vectors independent of Next we can take an integer [22], theguarantees smoothness that of Tr the Dirac operator in the admissible space (2.3) space. The full expectation value, including the gauge field sector, is thus given by v and 3. Global existence of the fermionIn this integration section, measure forthe real gauge-invariant representations, fermion we measure will globally show andfiguration smoothly that space over it (or the is more gauge-field possiblesible precisely, con- to within space). construct each The connected underlying component symplectic in structure the plays admis- the key role in this. vectors d field and it thus has no physical relevance. topological sector. The number of integration variables in D[ be done by the Gram-Schmidt method as JHEP10(2000)039 . is iθ − )= e Q 1 (3.1) (3.2) [1, 3] always − l 2 ]—we η Q and that ψ measures L ,v , i.e. 1 j are related ∗ v − l j 2 Q e v v within a local ( , = η j δ v +1 1 l and − j,k P j J has the eigenvalue δ i v 0 Q ξ and (3.1) hold. These − J ,k jk )] = δ ). The reason is that the l +1 2 j and we need to change the ,then v δ , ξ η information, but nevertheless 2 x, µ ) iθ )= ( u = consistency of the measure. e ,δ k l satisfies U 2 jk = ,v x, µ v j ( local ξ Q ), we have shown that it is always v U Q ) global . The proof of this important fact is , ], where the measure term ,( x )+( j x, µ , this implies that the eigenvalues of Q ψ ( 1 ( ξ v ,J µ − − l ) U D[ 5 2 = and the associated measure for η x aη v = ( unique 9 L η j k 00 i v v ξ ,δ − )= − also satisfies (3.1). Since 1 jk . This implies that the symplectic measure is ˆ =1 P − 1 J j l − 2 e v Q l x, µ ). For the symplectic measure, the measure term v ∗ 2 ( can be characterized by j ]= v [( )= v l U 1 j ψ continued to other gauge-field configurations, at least η x η v . − ( δ P D[ ∗ j ,δ J i B is an eigenvector of j η v − δ v 1 ξ such that . ( = − = j iθ = j η B − 1 .If v P T e ∗ L − i l smoothly ξ J 0 2 1 )= v − = and x J happens to have no component of .Thisbasis ( l η 0 = j iθ 0 2 2 ≡ e v ’s. But such a situation cannot occur for sufficiently close neighbors L v v l 0 j 2 are linearly independent and ξ − u v 0 =1and ξ = ). J 1 † − and J l are identical. 2 x, µ ξ j v ( Under the infinitesimal variation of the gauge field Now, cover the gauge-field configuration space by a collection of local coordinate For a fixed gauge-field configuration Therefore, it is always possible to construct a smooth basis e We define v changes smoothly. The smoothness of the construction breaks down only when, moreover can be 9 U j j symplectic. Namely, we have det the symplectic condition (3.1) guarantees the is defined by 0, because simple: assume a different basis patch in the gauge-field configurationthat, space as such that long (3.1) ascall holds. condition this Now (3.1) we the is can symplectic show satisfied, measure the — corresponding is measure D[ labelling of for where v identically vanishes, by (2.4), (3.1) implies that the unitary matrix patches. Within eachdescribed patch, above. we can Inthat an construct in overlap another the patch of are smooth not twoare symplectic necessarily patches, identical, the measure the same. since both as basis However, corresponding areshows vectors symplectic, that in and one it the patch symplecticconfiguration is and measure space. always is The possible unique. importantmeasure to This point within is define that a a the local smooth construction patch of measure requires the over symplectic only the the gauge-field and of the measure changes as for example, within a sufficiently small local patch containing above construction is purely algebraic andv when the gauge field is continuously varied, possible to construct come in pairs as Since JHEP10(2000)039 1 − 11 − ˆ has P B η (3.3) (4.1) − L =0is ˆ P and η By using B .Apar- L under the D j is as follows: U(1), † 10 = u η Q × D L T − = ˆ P j . u ) does not vanish and ln det F − − η ˆ ˆ . P P ) in (3.2). iδ ) )and hOi x − ] x ( ω − η ( + ( ω ] in (2.1), we have as the = 0) is gauge-invariant. j η L P µ u i R η L )( ,D L − ω − ) ( = −∇ ) ω )= R = η ( − x satisfying = 0 for the symplectic measure. e ˆ ( L R P j T j )= η ) u u x L − ω ( =[ − ( µ ˆ + P η is the covariant difference operator. R D P to eliminate the combination inside the square η , )( δ /a η 1 ω )] − L 6 ( , x ( B ) R ω + x is provided by eigenvectors of the hermitean ( − j j 1 u BP has been known non-perturbatively on finite lattices [1]. measure with vanishing measure term u 2 j − +[Tr has been known, but only to all orders in the perturbation ) η λ = F L η L any x, µ T + ( Oi )= P η U x ) δ ( h ˆ µ representations, the quantity Tr j a : = , + 2 Du F ) F † x ( D D ω 5 hOi ) complex hOi γ η δ x, µ ( =( U D † )=[ D x ( ω µ It remains to be shown that the symplectic measure is gauge-invariant. The ∇ For anomaly-free 10 11 identical to the symplectic measure up to ainfinitesimal constant gauge phase. transformation is given by the gauge covariance of the Dirac operator In the quantity inchange square of brackets, fermion the variablesthemselves first and change under term the (3.2). comes second We from term showed the that from jacobian the of fact the that basis vectors On the other hand, noting change of basis vectors (2.4), ticularly convenient complete set commute. For later comparison with the Majorana formulation, we need to know 4. Fermion expectation values In this section, we explicitlyplectic compute measure. the expectationbe value As (2.1) constructed shown by using starting in the with the sym- any previous complete section, set the symplectic measure can This establishes the existence oftopological a sector; globally consistent there gauge-invariant is measure no in any global obstruction for real representations. for real representations,expectation we values see of that gaugesymplectic invariant the measure operators (or first more are generally term always any measure gauge-invariant identically with and vanishes. the Namely, independent of the gauge field withinIncidentally, a since connected the component measure in the term admissible transforms space. as (This choice is analogous to thatcontinuum in theory the treatment [27].) of covariant gauge These anomalies two in conditions the are consistent because operator gauge variation of theory on the infinite lattice [4, 6]. For the representation in the electroweak SU(2) brackets is the aforementioned problemcohomological of the techniques local [7, gauge 3, anomaly. 4]. This The problem current can status be of studied our by knowledge concerning For general compact gauge groups, been known non-perturbatively at least on the infinite lattice [5]. the way to (and whether it is possible to) choose when the gauge group is U(1), such JHEP10(2000)039 . . j ]. = = 2. − ± u ψ / /a =0 j N 2 ± 0 )] (4.2) v (4.4) (4.3) n ϕ is the − ± , λ as ± +(1 j n − v 6= P that sat- . (III) n ± 2 j + n n e + ϕ j λ )=0.In + [ λ 6=0: v 0 n − N n − j = ˆ give the same P + ,ϕ λ . + 2 n n ∗ n ) − including ; the eigenvalues ϕ 5 ϕ for n 1 N Tr 1 − λ j , . 6=0and Dγ as the corresponding + v − ( B − n j + ± n ··· v λ to = , N gives rise to the solution ( j 2 0 n = v , (i) . because ( n ϕ − † e 0 ϕ n 13 ,wethusobtain , =1 j ϕ > v and .Thenumber DD j j and ± n v =0 to n is [Tr 1 ϕ n ,n , .DenotingΨ − j j j ϕ : and the total number is u u v as N /a + 2 ,λ . WedenotethenumberofΨ ± 0 − = ) Ψ ± :real ± ϕ x + − n ( + 7 4 has the eigenvalue ˆ = † P N / P span a complete set. In this way, we have: j 2 n n D =Ψ from category (i). But since v + † n j λ λ 2 v and the total number is 2 ϕ ± − a j ,λ 1 n Ψ − 0 λ ) /a − ∓ ϕ x . Below we will use this particular basis to compute . (ii) representations, all the eigenvalues − ˆ ( 1 P − n jk n ˆ + λ P )= δ doubly-degenerate 6=4 q ϕ n = x / − n real ( . This mapping gives rise to the symplectic structure 2 j n j λ ± λ jk , because ϕ )= v δ Ψ k − and 2) is linearly independent with ˆ ± P are even numbers. )= ,v n .Thisisgivenby P j 0 n x 2 ± λ v are classified into three categories: ( ϕ )= n n aD/ † k =0.For n /a v 12 ϕ 5 − , is hermitean. γ 6=0and are moreover Dϕ † j =4 and 5 v (1 D γ 2 j so constructed satisfies 2 j ,and 5 5 ± /a . We denote the number of λ n γ λ γ j 2 λ ± 0 N v ± ϕ ≡ ] = 0, only one linear combination of and ( n =0.Thisisgivenby = 0. One can choose the eigenvectors with definite chiralities as n = j with with = ϕ 2 j e ϕ n n j λ j /λ 2) λ λ ± 0 u Following the previous construction from The expectation value (2.1) also depends on how we choose the phase of D[ Once having obtained the solution of (4.2), we can obtain all the solutions of (4.1) † j u / Since it is simple to prove the following statements, we do not give the detailed proof. Note that ϕ v n † 13 12 ± ˆ The eigenvectors We fix this phase by the following natural mapping from Then aλ of (4.1). Thus the total number of this type of and Note that D P eigenvalue thus come in pairs as starting with Tr (iii) isfy (4.1), (3.1) and ( particular, (II) the expectation value (2.1).dependent Recall, of however, which that kindcondition the (3.1) of fermion is basis measure satisfied. vectors itself is are in- employed, as long as the symplectic analytic index on the latticeadmissible [22], space. which is For constantcan the in a show number connected the of component index eigenvectors in relation of the [28] the latter two categories, one by simply multiplying with some details concerningauxiliary the problem eigenvalue problem (4.1). For this, we consider the (I) eigenvectors, one has JHEP10(2000)039 j F is v → ,as 0 /a F (4.6) (4.5) (4.7) D hOi .For > i 5 k 6=2 n 1 γ c n λ h λ / are doubly- det F Q , i 2 and √ D ) / ) 5 ± j y γ − c ( (i.e. one factor n 2 T = / + n ) ψ + F λ ) − i n n ( x 1 h ( + m ) + ψ 2 . n h ) / ( ) vanishes when there ex- + ) in (4.6) due to one pair x, y 0, N ( F x, y i  im T x, y m ( − ( 2 1 ( ˆ → + 2 h P = 0, a mismatch between + / 1 m P + P † 2 j − 0, m 2 N + λ D B as the basis vectors for the zero 2 m 2 2 → 4  † a 2 + 0 m m +  m ) etc., although we do not write them m ϕ ) † im y + + 2 . im 1 ( + † + j = m − D D v 2 n DD † † ) 4 + a + 8 x D D B has a definite sign, up to a proportionality P (  2 n † T j − − + λ F + 0 − ˆ ˆ v ( P P P i  j ϕ 1 0 /a i h = = = ) > P 6=2 2 n Y n λ F F F λ m counting the double degeneracy of i i i in (4.2). In this expression, the product times the Wick contractions of fermion fields. The ) ) ) )= 2 / y y y n + . For the zero modes F ( ( ( F )] j λ i F F 2 n 0, this expression may be interpreted as i T v − ψ ψ i i x, y 1 , after a careful calculation using the above relations, we λ 1 n ) ) ( h 1 1 ψ ( h F → − h h ) x x − i In the massless theory ( ( − + ˆ x without 1 P and it thus precisely cancels one h n ( T h ψ m ( 15 h ψ ψ . − 0 /a h h > − /m also for 6=2 n Y 1 n n λ [Tr 1 k λ − i − v B = = + ) jk y J n F ( . The total number of these is i ). We have used the index relation (4.3) in deriving the second line. j = 1 −† 0 n v h λ ϕ B ) may occur and we can take ∗ j x j v ( v 14 We have completely fixed the phase ambiguity for the measure in (2.1). What The expression (4.6) holds for any topological sector. Interestingly, in the mas- − 0 This fact may be of interest from the viewpoint of numerical simulations. In the massless limit ≡ 15 14 iϕ 0 j in terms of the eigenvalues − constant that dependscombination only on which topological sector is concerned through the ists a zero mode, as should be the case. The general fermion expectation value the partition function for each modes in down explicitly. For example, in the massless limit It is easy to express thesetions basic in contractions in (4.2) terms by of noting the eigenvalues and eigenfunc- understood to be taken is computed as usual by and v basic contractions are given by remains to be done is simply the Grassmann integrals with respect to have sive theory, the partition function degenerate, even if some ofthere the is eigenvalues cross no zero ambiguity accordingeigenvalues in to that a the cross deformation sign zero ofdoes the of [13]. not gauge the appear field, This square for explains real root representations (for [10] from the the because vacuum viewpoint it of sector) is the why spectral always the flow. an Witten even anomaly number of naively expected for the Weyl fermion in a real representation. Since eigenvalues of JHEP10(2000)039 , is 2 / ) ]= − CD − χ (5.2) (5.1) n n − + + = n ( 2 T / ) ) m 2 − / n , + CD +  N ) +  n x 2 ( ( ) . With this choice, m χ n 5 This property of the im ϕ + − 2 Cγ ( 4 ) a 16 2 x /  ( − ) T 2 N  m imχ + im 2 1 2 n − λ ( 0 /a 2 a > )+ 6=2 n  Y x n ) and (5.2) gives the precise meaning of spinor field. Note that ( λ 2 ( 2 λ 5 / / + ) 9 − N n  CDχ − ) imCγ + im x n , where the fermion integration measure is de- ( ( +  T − χ 2 a im 2 ’s are certain orthonormal basis vectors) and D[ 1 CD appear symmetrically in the first expression, because n  − Majorana F unconstrained − ± ϕ  S Pf( ( x ) 2 a N − X 2 e n  4 ∝ b . The important difference from the Weyl formulation is m 2 a ) O / ) ] x + Tr 1 − ( = χ being consistent with the Fermi statistics and that the mass b n 2 n n d − 5 λ ϕ D[ ( + n Majorana F R n 0 /a ( ··· i Cγ > P 1 − 2 6=2 − n i = h Y b Majorana n F λ λ d − S 1 = )= b x = = d T ( ) is a four-component χ 5 ≡ Majorana F χ n Cγ b We can take the eigenvectors in (4.2) as the basis vectors From (5.1), the fermion partition function in the Majorana formulation is given This is analogous to the situation for the Dirac fermion in lattice QCD in which one usually hOi d Majorana F i 16 n 1 h Majorana formulation has ansection. interesting implication, as we will discuss in the next of the left-right-symmetric treatment in the Majorana formulation. by the pfaffian where and ( where from the first line to the second line we have used (4.3) and the fact that we obtain as the fermion partition function term is gauge-invariant for realby representations. The expectation value is then given an even number. Note that Q As noted inories, the introduction, the in Weylfermion. four-dimensional fermion continuum in Thus (unregularized) it the is the- formulation real of in which interest representation the to is left-right seeimplementation chiralities equivalent of are how to treated the this asymmetrically. Majorana equivalence the The (left-right is Majorana lattice symmetric) realized fermion in would the be present given by 5. Matching to the Majorana formulation of left-handed zero modes.logical In sector this is way, obtained anyabove by fermion combining expressions, expectation (4.6) value expectation and ingauge-invariant. (4.7). values any Note of topo- that, gauge-invariant according operators to are the manifestly fined by that the Majorana formulation can beical set up sector, without because referring to the a number particularthe topolog- of above integration variables definition is is always the uniform same. for Namely, all topological sectors. never worries about the relative weight for the fermion measure in different topological sectors. JHEP10(2000)039 − (5.5) (5.3) (5.4) . 2. The # (assum- # Weyl F 1 x,y 2 iE − D/ / δ 5 4 T + C 1 − 5 N aγ − ) a γ 2 ) + C − ) /a . y − ( x, y CP ˆ (4 P ( ψ 2 a 2 n − Weyl F λ + ˆ i P − + 1 + 2 ) a 0 /a + h > y + P 2 P 6=2 ( n / ) − P n λ ) T T λ Weyl F T − T and, when the overlap-Dirac − i C ψ n ˆ Q C 1 x, y P − ih ( h ) + 1 − n x − CD ( ( =  C , T ) Pf Majorana F 5 ψ i Majorana F In particular, they lead to the same Majorana F im 1 x,y ) i ∝ i δ − ) y ) 4 − ( y y 17 imγ C Majonara F − ( ( T i Majonara F 2 a Majonara F Majorana F a T i T 1 χ i − − i 1 10  h ) 1 χ 1 χ 1 † 4 − ) h ) h ) x h / x ( x Majorana F C D x ( ( i ( 5 ( χ Tr 1 1 ψ h χ 2 γ χ h h h 1) C m Dh − − im T ˆ + − , this coincides with the expression in [18], which ˆ P P + ( P im − 1 " D 2 1 " − ) im − /a /a D = 2 im − /a 2 5 im /a 2 γ − /a − /a 2 ( ) decays exponentially with a fixed range in the lattice units [23], this − /a 2 2 2 /a /a = = x, y 2 2 − ( Majorana F − i ˆ = = = P 1 h Majorana F Weyl F Weyl F Weyl F i i i i ) ) ) ) y y y y ( ( ( ( Weyl F Majonara F Weyl F Weyl F i T T i ψ ψ i i 1 ) ) 1 χ 1 1 ψ h h ) h h x ) x ( ( x x Comparing (5.2) and (4.6), we find ( Since the kernel ( T ψ χ h 17 ψ ψ h h h Therefore, with these rulesthe (5.4) two and formulations, (5.5), up the to expectation contact terms. values are identical in Namely, two formulations match up toon a the proportionality topological constant that sector.two depends formulations only If are one therefore completely isfind equivalent. concerned For with the a basic particular contraction, topological we sector, ing there is nofrom zero the mode), viewpoint which of numerical manifestlypartition simulations has function [13, then a 14, allows definite 16, a 17,the sign. statistical 18], overlap-Dirac weight because This interpretation. operator the is has This fermion important propertydure been with from shown the [18] domain byhave wall appealing shown fermion the to with same the finite property limiting five-dimensionalalone. by proce- using separation. general Here properties we of the GW Dirac operator where, in deriving the last expression, we have noted 1 = physical amplitudes with that matching rule. relation in the opposite direction is given by this pfaffian. In the massless limit, can effectively be regarded as a contact term in the continuum limit. operator [21] isis employed based as on afive-dimensional factorization separation) property or of thetheories. the overlap In domain [29] this limit, wall fermion (5.2) [15, determinant reduces 14] in to (with vector-like the infinite JHEP10(2000)039 . ). j v M ,we ( (6.1) (6.2) (6.3) ∗ N R 18 normalization , .Now,asalready ∗ R R ⊕ by using (4.6) and (4.7) as it coincides with the ) in (2.5)) for the Weyl ,R Majorana F ,R R F , M . ( F is specified by the basis vec- Oi F ∗ , ,R ∗ N hOi F ∗ . hOi R for all topological sectors. Thus hO hOi R + − ⊕ iϑ P ⊕ n hOi e T ,R ∗ refer to the numbers of zero modes − 2 F R / C + ,R ) ) n ± F − x i Oi (  n . With this choice of measure for n ∗ ∗ ∗ T 2 a − uniform . This measure is symplectic with respect χ + R  T 11 hO hO n 4 ) ( / ∗ , we may use also the Majorana formulation. l  ∗ = = v →− 2 a 1 Tr 1 ) R 2 − | x  representation. For complex representations, the ( 1) ⊕ ,R ψ ,B F − → representation ( R F − naturally provides a consistent gauge-invariant measure |hOi =(0 ∗ )and j = real l x v hOi representation, which is specified by the basis vectors 2 ( 2 1 complex | χ V − In this expression, and, as naively expected [12], − ,R ˆ B ∗ P F and we can thus apply the previous arguments. In particu- ], this does not affect the following argument for the of the proportionality constant in this expression depends on a way we 19 ,R and χ ∗ → j F i ) V T |hOi ∗ 1 complex x ( 0) − Weyl fermion in the complex representation phase ψ , hO l B v  = 0. 01 10 =( ,R  ∗ F 1 → − original l ≡ 2 hOi m 0 V j Suppose that we have a consistent gauge-invariant measure for the Weyl fermion For the real representation Although the Eq. (5.5) shows that the substitution rule from the Weyl formulation to the Majorana formu- V 18 19 where the measure for the up to contact terms. with specified the phase of D[ tors lar, (6.1) shows that we can compute the modulus of to of the From (6.1), (5.3) and (5.5), we know that for a fixed topological sector: Then the set of vectors normalization of the Majoranamay formulation define for the relative all weight topological for sectors. a topological Namely, sector we as have for the complex conjugate representation We have seen that therethe is Majorana a complete formulation matchingpossible for between implication real the of Weyl representations. formulation thismeasure and matching in In for different this the topological section, relative sectors normalization we (the of present the factor a fermion 6. Relative normalization for different topological sectors fermion belonging to the mass term breaks the gaugetheory. symmetry. We thus restrict our problem to the massless belonging to the lation is given by emphasized, the Majorana formulationit is is quite natural to adjust the normalization of JHEP10(2000)039 , , ]. (6.4) Nucl. , , which de- M F , hOi ). On the other hep-lat/9909150 M F M andez, Karl Jansen ( N hOi ) M ( iϑ ]. e (2000) 34 [ . Therefore, the Weyl formu- 2 / ) /k − ]. n − + n Chiral gauge theories on the lattice with (  =2. ]; 2 a was determined from the matching with k  2 (Proc. Suppl.) / 12 G ) S hep-lat/9901012 − − 83 n e − ] + n (note that the mass dimension of the Grassmann U ( hep-lat/9811032 − D[ n ) cannot be fixed from the present argument. As- M has a definite mass dimension, the dimensionful fac- Z − M hep-lat/9904009 ( (1999) 1147 [ O + Nucl. Phys. ϑ M X n , , the number 2 changes to 2 1 Z D (1999) 295 [ 101 5 )= 2). This is a natural requirement for = − / ˆ P compensates changes of the mass dimension of (2000) 162 [ 2 Weyl fermions on the lattice and the nonabelian gauge anomaly kaDγ Abelian chiral gauge theories on the lattice with exact gauge invariance − is 1 hOi / B 549 ) Gauge invariant effective action in abelian chiral gauge theory on the lattice + − j = c n P − D + 5 B 568 n uscher’s gauge-invariant lattice formulation. We hope that we clearly illus- uscher for helpful discussions and suggestions, which quite enriched the γ uscher, uscher, ( ) + 5 /a Nucl. Phys. Prog. Theor. Phys. Phys. exact gauge invariance The question raised by Taku Izubuchi many years ago initially motivated the Dγ [1] M. L¨ [2] H. Suzuki, [3] M. L¨ hand, the relative normalization 2 integration d where the relativesuming phase that the operator tor (1 pends on Tr( the Majorana formulation.as If one chooses the normalization of the GW relation lation will automatically givethe rise normalization to of the the natural Dirac relative operator normalization as by choosing References contents of this paper.where this I work am was grateful done, to especially members Patricia of Ball, the Pilar CERN Hern´ Theory Division, The real representation,anomaly, provides owing an interesting to exampletities with its in which simplicity L¨ one can with worktrated out some regard all global the issues to quan- in theimplication the formulation of with local the this present simple analysis gauge example.provides is a An that natural interesting the normalization of matching the tological fermion-integration the measure sectors. 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