conrm that at least the chiral eterminant can be ene on the lattice using the overlap formalism. The overlap formalism has been applie by a number of

Transcription

1 The Chiral Dirac Determinant Accoring to the Overlap Formalism Per Ernstrom an Ansar Fayyazuin NORDITA, Blegamsvej 7, DK-00 Copenhagen, Denmark Abstract The chiral Dirac eterminant is calculate using the overlap formalism of Narayanan an Neuberger. We compare the real an imaginary parts of the eterminant with the continuum result for perturbative gauge el backgrouns an show that they are ientical. Thus we n that the overlap formalism passes a crucial test. Norita 95/80 November, 995 Lattice regularization of el theories is the only known non-perturbative regularization available to us. Chiral gauge theories have elue a non-perturbative regularization for reasons summarize by the Nielsen-Ninomiya [, ] theorem which states that there exists no iscretization of the chiral Dirac operator which simultaneously preserves a number of esirable physical properties. This is an unfortunate state of aairs since, at least at low-energies, the Weinberg-Salam moel escribes the physics of the worl we live in, an this moel involves chiral couplings of fermions to gauge els. Recently attempts have been mae to evae the theorem of Nielsen an Ninomiya in various ways (see [, 3] for a recent review of progress in this irection.) We will be concerne with the approach of Narayanan an Neuberger[4] who, inspire by an iea of Kaplan's[5], have propose a new way of calculating chiral uantities on the lattice. They evae the Nielsen-Ninomiya theorem by stuying an auxiliary problem in one imension higher. Quantities in this auxiliary problem can then be relate to the lower imensional theory by taking certain limits. Thus the conclusions of the no-go theorem are avoie by formulating a problem which ostensibly has nothing to o with the original problem an in fact is formulate in o imensions where chirality is not an applicable concept. In this letter we calculate the eterminant of the chiral Dirac operator in 4 imensions using the recipe of Narayanan an Neuberger which has been bbe the \overlap formalism". First we evaluate the molus of the eterminant an show that it reproces correctly the continuum result. We then evaluate the phase of the eterminant an compare our result with the continuum result of Alvarez-Gaume, Della Pietra an Della Pietra[6] (similar expressions have also been erive by [7, 8]. We n that the results of the two approaches are ientical. While calculating the imaginary part of the effective action we always work with perturbative gauge els since this is the assumption uner which the continuum results have been obtaine. Our results

2 conrm that at least the chiral eterminant can be ene on the lattice using the overlap formalism. The overlap formalism has been applie by a number of authors to various problems involving chiral an non-chiral fermions. Ranjbar-Daemi an Strathee have calculate chiral anomalies in an 4 imensions, the gravitational anomaly in imensions an the vacuum polarization in 4 imensions [9, 0, 3, ]. They have also calculate the two point functions for chiral fermions [9] an verie anomaly cancellation in the stanar moel using this formalism. Ranjbar-Daemi an Fosco calculate the eterminant of the chiral Dirac operator in a constant backgroun gauge el with non-trivial holonomy on the two imensional torus, verifying that the continuum result is reproce incluing the holomorphic anomaly []. Narayanan an Neuberger have applie their formalism to the twiste chiral Dirac operator an conrme numerically the continuum result [4]. Narayanan, Neuberger an Vranas [5] applie the overlap formalism to the Schwinger moel an obtaine results consistent with the continuum exact solution. This last piece of work is particularly interesting in that the gauge els involve in that calculation are topologically non-trivial an therefore involve zero moes of the Dirac operator. While work on the present project was in progress we receive [6] where the phase of the chiral eterminant is calculate for omain wall fermions. The authors suggest that their results also apply to the overlap formalism. The overlap formalism expresses chiral uantities in terms of certain objects in an auxiliary problem in one imension higher. Specically, one consiers two ve imensional Hamiltonians (we use the simpler notation of Ranjbar-Daemi an Strathee evelope in [9] an [3]) H = Z 4 x y (x) 5? D= (x) () where > 0 is a mass for the ve imensional fermions. Notice that the two Hamiltonians ier only by the sign of the mass term. The Dirac vacua for the two Hamiltonians are enote by j A > which are Slater eterminants of the non-positive eigenvalue states of the rst uantize Dirac Hamiltonians. The overlap formalism states that the chiral Dirac eterminant is given by the following expression: et[ ( + h+ j A+ihA+ j A?ihA? j?i 5)(D= )] = lim! h+ j?i j h+ j A+ih? j A?i j where j > are the Dirac vacua of the problem with vanishing gauge els. Before proceeing to the calculation we comment on the regularization procere. We assume that the ve imensional overlap problem can be regularize on the lattice. We will never explicitly ene a lattice regularization but will assume its existence an other formal properties of the lattice regularize operator which must be ientical with the continuum problem. The Dirac operator as we will use it will always be a nite imensional matrix an we will take =a + jjajj where jjajj is the supremum of the expectation value of the For our calculation of the molus of the eterminant it is sucient to take jjajj ()

3 gauge el an a is the lattice spacing, ensuring that the eigenvalues of the Dirac operator are small compare to the mass. There are two large scales in the problem: an the inverse of the lattice spacing =a. Both are large but the relevant limit to reproce the continuum result is the one in which =a. However, one coul imagine various limits controlle by the imensionless parameter a. We will have nothing to say about this but we hope to return to this issue in the future (see, however, the iscussion in [3] ). It woul be very interesting to characterize any gauge non-invariant terms, which we neglect in the present work, \suppresse" by a an others suppresse by a an =. This may she light on the continuum limit of the lattice regularize overlap formalism. The four imensional Dirac operator anti-commutes with 5, this allows one to pair the non-zero eigenmoes of the operator as follows: D= (A) j (A) = i j (A) j (A) ; (3) D= (A) 5 j (A) =?i j (A) 5 j (A) : (4) We aopt the notation that the j are positive eigenvalue moes of?id= an j enotes a positive eigenvalue. We assume rst that there are no zero moes. Then the eigenstates f j ; 5 j g of the Dirac operator form a complete basis. We will always assume that j (0) an j (A) are smoothly relate to each other by an interpolating gauge el A t between the gauge congurations 0 an A such that the Dirac operator oes not evelop a zero moe anywhere along the interpolation. The rst uantize hamiltonians: h = 5? D= (5) commute with D= an allow one to iagonalize h an D= simultaneously. We can express the eigenstates of h as linear combinations of a positive eigenstate of D= an its negative eigenvalue pair. This has the virtue that the eigenstates of h will be linear combinations of j ; 5 j an the epenence will occur only in the coecients multiplying the eigenstates of D=. Using the usual notation enoting positive (negative) eigenstates of the hamiltonians h as u (v ) we get the following eigenstates: 0 v ;j (A) = u ;j (A) = p + i j (A) j (A) + 5 A j (A) ; i j (A) j (A) + 5 A j (A) : (6) These wave functions satisfy: h v ;j =? h u ;j = + j + v ;j ; j + u ;j : (7) After submitting this work we receive a paper by Ranjbar-Daemi an Strathee where this uestion is taken up (eprint archive: hep-th/95) 3

4 The Dirac vacua are then given by the Slater eterminants of the negative energy states. We have normalize the above wave functions in such a way that if we have an inner proct such that ( j (A) ; k (A)) = ( 5 j (A) ; 5 k (A)) = jk ; ( j (A) ; 5 k (A)) = 0: (8) then the v ; u are orthonormal with respect to the same inner proct. Now we will calculate ha+ j A?i. The states j A > are Slater eterminants of the v, assuming for the moment that D= has no zero moes, ha+ j A?i = etm; M jk = (v +;j ; v?;k ): (9) We can now evaluate M jk using euations ( 6) an ( 8). We n M jk = jk j (A) j (A) + i ; etm = Y j j (A) j (A) + i : (0) Thus ha+ j A?i lim! h+ j?i Y j (A) j (0) + i = lim! j j (A) + i j (0) = Y j = vu u j (A) j (0) t etd= (A) et@= () : () The last euality follows from recalling that the j are the positive eigenvalues of D=. Now we woul like to show that if D= has zero moes then ha+ j A?i = 0. To emonstrate this we ivie the zero moes of D= into positive (L) an negative (R) \chirality" (i.e. with respect to 5 ) moes. Then enoting by L(R);j the zero moes of D= we see that they are eigenstates 3 of h : h L;j = L;j ; h R;j = R;j : (3) Therefore, the right hane zero moes will be in the j A+i vacuum but will not appear in j A?i. The converse is true for the left hane zero moes. Using the orthogonality of the left an right hane moes then proves that ha+ j A?i = 0. Of course, if the number of left an right hane zero moes 3 Note that the inex j oes not necessarily run over the same number of values for the left an right hane wave functions, this number can be ierent if the gauge el carries a non-zero instanton number. 4

5 is not the same then ha+ j A?i = 0 for an aitional reason than the one just state, namely, there is a mismatch in the number of states in the two vacua. So far we have evaluate ha+ j A?i=h+ j?i for arbitrary gauge congurations. Since the remaining part of the Dirac eterminant ( ) is a phase while ha+ j A?i=h+ j?i is a real non-negative number we have evaluate the magnitue of the chiral Dirac eterminant an foun that it is precisely as it shoul be. We turn now to the phase of the Dirac eterminant which is, in a sense, at the heart of the matter since all the information about chirality is store in this phase. The magnitue is merely the suare root of the full Dirac eterminant with vector couplings. We are intereste in calculating the phase of the eterminant in backgroun congurations for which there are continuum results available for comparison. The phase of the eterminant was calculate by Alvarez-Gaume, Della-Pietra an Della-Pietra for perturbative backgroun gauge els for which there are no zero moes of the Dirac operator. They foun that the phase can be written as: h ^At; i Im ln et = (0)? Q 0 5 ^At; : (4) Where Q 0 5 is the ve imensional Chern-Simons form an ^At;u is a two parameter extension of the four imensional gauge el such that ^At;u = 0 for? t?t, ^At;u smoothly interpolates between 0 an A u for?t < t < T an nally ^At;u = A u for T < t <. While A u is an interpolating el between 0 (for u = 0) an A (for u = ). A is the four imensional gauge el appearing in the Dirac operator whose eterminant we wish to calculate. The object u (0) is the so-calle eta-invariant associate with the ve imensional Dirac operator H u = i t + id= ^At;u. Crucial to the erivation of this result is the ientity erive in [6]: u (0) = Q0 5 + tr 5! D= (A u ) D=? (A u )? D=?= (A u ) =M (5) The M appearing on the right han sie is a Pauli-Villars mass regularizing the expression, an a limit where it is taken to innity is implicit. If one takes this limit the last term on the right han sie becomes: lim M! tr 5 D= (A u )! D=? (A u )? D=?= (A u ) =M! = tr D= (A u ) 5 D=? (A u ) : (6) This expression, without the Pauli-Villars mass, has appeare in the physics literature previously in a paper by Niemi an Semeno [7]. Alternatively, one coul ene a lattice regularization of the operators appearing in the trace an then remove the Pauli-Villars regulator. In either case we nee to take this limit to be able to compare with our calculation which has no Pauli-Villars regulator 5

6 but a lattice regulator instea. Finally, we can evaluate the trace over the basis f j (A u ) ; 5 j (A u )g to get Where an u (0)? Q 0 5 ^At;u = tr 5 = X k! D= (A u ) D=? (A u ) k (A u ) ; 5 k (A u ) The phase of the eterminant in the overlap formalism is given by : (7) Im lnh+ j A+ihA? j?i = Im (ln etm + + ln etm? ) : (8) h+ j A+i = etm + ; M +jk = (v +j (0) ; v +k (A)) =? ha? j?i = etm? j (0) + i + (0) + j (0) + j (0) + + M?jk = (v?j (A) ; v?k (0)) = k (A) + k i k (A) + k (A) + (A) + k A (j (0) ; k (A)) A (j (0) ; 5 k (A)) 5 ; i j (A) +?i + k (0) + A (j j (A) + (A) ; k (0)) (0) i j (A) + j (A) + + i k (0) + k (0) + A (j (A) ; 5 k (0)) 5 : (0) We now take the limit! an evaluate the eterminants in that limit. M +jk! ( j (0) ; (? 5 ) k (A)) ; M?jk! ( j (A) ; ( + 5 ) k (0)) : () To compare the overlap metho with the continuum metho we consier a el A u interpolating between the congurations A 0 = 0 an A = A. Using the completeness of f j (0); 5 j (0)g It is easy to check that M are unitary matrices in the limit!. Thus: ln etm u = trm y u M u X = k (A u ) ; ( 5 ) k (A u ) k 3 3 : () 6

7 We arrive nally at the expression: lnh+ j A+ihA? j?i = (ln etm +u + ln etm?u ) X = k (A u ) ; 5 k (A u ) k : (3) Comparing with euation (7) we see that the continuum result for the phase of the chiral eterminant is reproce by the overlap formalism. Together with euation () we have s h+ j?i j h+ j A+ih? j A?i j = etd= (A) et@= h+ j A+ihA+ j A?ihA? j?i lim! e i( (0)?Q 0 5( ^A t;)) : (4) We conclue with a few comments. The overlap recipe for the chiral Dirac eterminant has passe an important test by reprocing the continuum result. What is most satisfying about this result is that while in the continuum the imaginary part of the eective action naively vanishes but is proce e to the regularization of the eterminant an survives the limit in which the regulator is remove, in the overlap formalism one can reproce this result inepenently of the specics of the regularization procere an any elicate limits. This is a conseuence of the fact that we are always working with a parity non-invariant system which has a non-vanishing imaginary part inepenently of the lattice regulator. In our erivation we have neglecte terms of orer = where is a typical eigenvalue of the Dirac operator. It woul be interesting to keep these terms an to see how various limiting proceres in which a! 0 an! can aect continuum limits. In any problem where more than one large scale is available one must eventually aress the uestion of their relative size. Acknowlegements We woul like to thank B. Kileng, A. Krasnitz, L. Karkkainen, an especially A. Kronfel for iscussions. We are grateful to M. Schmaltz for pointing out some misprints in an earlier version of this article. References [] H. B. Nielsen an M. Ninomiya, Nucl. Phys. B85 (98) 0, B93 (98) 73. Erratum, Nucl. Phys. B95 (98) 54. [] Y. Shamir, \Lattice Chiral Fermions", e-print archive: hep-lat/950903, an references therein. [3] R. Narayanan an H. Neuberger, \Progress in lattice chiral gauge theories", e-print archive: hep-lat/ ; H. Neuberger, \A Lecture on Chiral Fermions", RU-95-79, e-print archive: hep-lat/9500 an references therein. [4] R. Narayanan an H. Neuberger, Nucl. Phys. B443 (995) 305. [5] D. B. Kaplan, Phys. Lett. B88 (99)34. [6] L. Alvarez-Gaume, S. Della Pietra, V. Della Pietra, Phys. Lett. B66 (986) 77. 7

8 [7] A. Niemi an G. Semeno, Phys. Rev. Lett. 55 (985) 97. [8] R.D. Ball, H. Osborne, Phys. Lett. B65 (985) 40, R.D. Ball, Phys. Rep. 8 (989). [9] S. Ranjbar-Daemi an J. Strathee, Phys. Lett. B348 (995)543. [0] S. Ranjbar-Daemi an J. Strathee, Phys. Rev. D5 (995)667. [] C. D. Fosco an S. Ranjbar-Daemi, Phys. Lett. B354 (995)383. [] S. Ranjbar-Daemi an J. Strathee, e-print archive: hep-th/ [3] S. Ranjbar-Daemi an J. Strathee, Nucl. Phys. B443 (995) 386. [4] R. Narayanan an H. Neuberger, Phys. Lett. B348 (995) 549. [5] R. Narayanan, H. Neuberger an P. Vranas, Phys. Lett. B353 (995) 507; an e-print archive: hep-lat/ [6] D. B. Kaplan an M. Schmaltz, \Domain wall fermions an the - invariant", e-print archive: hep-th/