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Chiral symmetry and spectrum of Euclidean Dirac operator in QCD


BUTP-95/4

Chiral symmetry and spectrum of Euclidean Dirac operator in QCD 1

arXiv:hep-th/9503049v3 29 Nov 1995

A.V. Smilga Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, Bern CH-3012, Switzerland 2

March 1995

Abstract Some exact relations for the spectral density ρ(λ) of the Euclidean Dirac operator in QCD are derived. They follow directly from the chiral symmetry of the QCD lagrangian with massless quarks. New results are obtained both in thermodynamic limit when the Euclidean volume V is sent to in?nity and also in the theory de?ned in ?nite volume where the spectrum is discrete and a nontrivial information on ρ(λ) in the region λ ? 1/(| < q ?q >0 |V ) (the characteristic level spacing) can be obtained. These exact results should be confronted with ”experimental” numerical simulations on the lattices and in some particular models for QCD vacuum structure and may serve as a nontrivial test of the validity of these simulations.

1

Introduction

The notion of spectral density of the Euclidean Dirac operator in QCD has been brought into discussion some years ago in the pioneering paper by Banks and Casher [1]. They have got the famous formula ρ(0) = ?
1

1 <q ?q >0 π

(1.1)

Talk given at the International Conference on Chiral Dynamics in Hadrons and Nuclei, Seoul, February 1995. 2 Permanent adress: Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow 117259, Russia.

1

relating the infrared limit of the averaged spectral density with the quark condensate — the order parameter for the spontaneous chiral symmetry breaking. ρ(λ) is de?ned as 1 < ω (λ, A) >A (1.2) ρ(λ) = V where ω (λ, A) = δ (λ ? λ n ) (1.3)
n

is the microscopic spectral density of the Dirac operator in a given background ?eld A in a ?nite Euclidean volume V . The averaging is taken over all gluon ?elds with the standard Yang-Mills measure involving the determinant factor for Nf fermions with the common mass m. The combination V ρ(λ)dλ de?nes the average number of eigenvalues of Dirac operator in the interval (λ, λ + dλ). Let us recall how the Banks-Casher relation (1.1) (which holds in the thermodynamic limit V → ∞, with the mass m being kept small but ?xed) is derived. Treating the gauge ?eld Aa ? (x) as an external ?eld, the fermion Green’s function is given by SA (x, y ) =< q (x)? q (y ) > A =
n

un (x)u? n (y ) m ? iλn

(1.4)

where un (x) and λn are eigenfunctions and eigenvalues of the Euclidean Dirac operator: D un (x) = λn un (x) (1.5) Note that the spectrum of the theory enjoys the chiral symmetry: ψn → γ 5 ψn , λn → ?λn (1.6)

And, except for zero modes, the eigenfunctions occur in pairs with opposite eigenvalues. Setting x = y and integrating over x, the representation (1.4) therefore implies 1 V
V

dx < q (x)? q (y ) > A = ?

1 2m V λn >0 m2 + λ2 n

(1.7)

where the zero mode contributions have been dropped out (it is justi?ed in the thermodynamic limit. See Ref.[2] for details). In the limit V → ∞, the level spectrum becomes dense and we can trade the sum in the r.h.s. of Eq.(1.7) for the integral. We get ∞ ρ(λ) (1.8) <q ?q >0 = ?2m dλ 2 m + λ2 0 The integral (1.8) diverges at large λ. For λ ? ΛQCD the spectral density is not sensitive to gluon vacuum ?uctuations and behaves in the same way as for free fermions: Nc λ3 ρf ree (λ) ? (1.9) 4π 2 2

This perturbative quadratically divergent piece in < q ?q >0 is proportional to the quark mass and is related just to the fact that mass terms in the lagrangian break the chiral symmetry explicitly. To get the truly non-perturbative quark condensate which is the order parameter of the spontaneous chiral symmetry breaking, this perturbative divergent part should be subtracted. As a result, the quark condensate is related to the region of small λ: λ ? m ? ΛQCD . It is not di?cult to see that the ?nite mass-independent contribution appears if ρ(0) = 0 and the relation (1.1) holds.

2

Thermodynamic limit: λ = 0.

In this section, we derive a new formula describing the behavior of the spectral density ρ(λ) in the region m ? λ ? ΛQCD in the thermodynamic limit V → ∞ [3]. To this end, let us consider the 2-point correlator K ab = d4 x d4 y < 0|S a (x)S b (y )|0 > (2.1)

with S a = q ?ta q where ta is the generator of the ?avour SU (Nf ) group. The correlator K ab can be evaluated in the same way as < q ?q >0 . First, ?x a particular gluon a background. As T r {t } = 0, only the connected part depicted in Fig.1a contributes, and one obtains
ab KA =?

d4 x

d4 y Tr{ta GA (x, y )tbGA (y, x)}

(2.2)

Substituting here the spectral decomposition for the Green’s function (1.4) and taking into account the chiral symmetry of the spectrum (1.6), we get
ab KA

m2 ? λ2 n = ?δ 2 2 2 λn >0 (m + λn )
ab

(2.3)

Rewriting the sum as an integral as in Eq.(1.8) and averaging over the gluon background, one gets ∞ ρ(λ)(m2 ? λ2 ) 1 ab K = ?δ ab dλ (2.4) V (m2 + λ2 )2 0 On the other hand, the correlator (2.1) of the colorless scalar currents can be evaluated by inserting the complete set of physical states. For small m, i.e. close to the chiral limit, a distinguished position among the latter belongs to the Goldstone states which appear due to spontaneous breaking of the chiral symmetry of the QCD lagrangian. Low energy properties of Goldstones are ?xed by chiral symmetry, and some exact calculations are possible. The chiral lagrangian has the form [4]
2 L = Fπ

1 Tr{(?? U ? )(?? U )} + B Re Tr{MU ? } + higher order terms 4 3

(2.5)

?2 where B = ? < q ?q >0 Fπ , U = exp{2iφa ta /Fπ }, and M is the quark mass matrix. 2 If M = diag(m, . . . , m), the lagrangian (2.5) describes Nf ? 1 (quasi-) Goldstone states with the common mass 2 Mφ = 2mB (2.6)

Consider now the graph in Fig.1b with 2-goldstone intermediate state contributing to the correlator (2.4) (obviously, one-goldstone state does not contribute since < 0|S a |φb >= 0). To calculate it, we need to know the vertex < 0|S a |φb φc >. It can be easily found from the generating functional involving external scalar sources. The latter is obtained substituting in the e?ective lagrangian (2.5) the mass matrix M by M + ua ta where ua is the source for the scalar current S a . In this way, one gets < 0|S a |φb φc >= Bdabc (2.7)

It is very important that the vertex is nonzero only for three or more ?avors. Now we can calculate the graph in Fig.1b absorbing its ultraviolet divergence into local counterterms contained in higher-order terms in the e?ective lagrangian (2.5) [4]. The result reads 2 B 2 (Nf ? 4) ab 1 ab 2 K =? δ ln(Mφ /?2 (2.8) hadr ) 2 V 32π Nf For massless goldstones, the graph in Fig.1b does exhibit a logarithmic infrared sin2 gularity re?ected in the factor ln Mφ on the r.h.s. of Eq.(2.8). The same circumstance makes our calculation self-consistent: since infrared singularity arises from the low momenta region, higher derivative terms in Eq.(2.5) can be neglected. Now, let us compare Eq.(2.4) with Eq.(2.8). Note ?rst of all that, in contrast to the integral in Eq.(1.8), the constant part ρ(0) does not contribute here at all:
∞ 0

ρ(0)(m2 ? λ2 ) dλ = 0 (m2 + λ2 )2

(2.9)

Thus, only the di?erence ρ(λ) ? ρ(0) is relevant. It is easy to see that, in order to 2 reproduce the singularity ∝ ln Mφ ∝ ln m, one should have ρ(λ) ? ρ(0) = Cλ (2.10)

2 at small λ. Comparing the coe?cients at the logarithms ln m and ln Mφ in Eq.(2.4) and Eq.(2.8), we arrive at the result 2 <q ?q >2 1 0 (Nf ? 4) | λ | + o (λ ) <q ?q >0 + 4 π 32π 2 Nf Fπ

ρ(λ) = ?

(2.11)

The structure |λ| appears due to the relation ρ(?λ) = ρ(λ) being implied by the chiral symmetry (1.6). Note again that this non-analyticity does not appear at Nf = 2. 4

Recently, the preliminary, not yet published data for the spectral density ρ(λ) calculated on the lattice with two dynamical fermion ?avours appeared [5] . The same quantity has been also evaluated in the instanton-antiinstanton liquid model for QCD vacuum con?gurations [6]. Results of these two numerical studies are completely different. Lattice calculation failed to reproduce the exact QCD relation (2.11) (they exhibit a large nonzero slope for ρ(λ) which should be absent for Nf = 2). Probably, it is due to the fact that , in the region of lattice parameters used in [5], the thermodynamic limit was not reached yet and the ?nite volume e?ects were still important [7]. Obviously, further studies in this direction are highly desirable. On the other hand, the calculations in the instanton model agree well with theoretical predictions. The results for the spectral density obtained in [6] for di?erent number of light quark ?avors are presented in Fig.2. Before comparing them with the theory, two remarks are in order. 1. Only non-perturbative instanton-driven part of spectral density has been determined in [6]. So, the perturbative e?ects which give the dominant contribution to the spectral density at large λ were not taken into account, and the comparison with theory makes sense only in the small λ region. 2. The abrubt fallo? and vanishing of ρ(λ) at λ = 0 as measured in [6] is the ?nite volume e?ect. The comparison should be done in the region of not yet too small λ. From the graphs in Fig.2, one sees that, for Nf = 2, the spectral density is practically ?at, while for Nf = 3 it rises with a nonzero slope. This is exactly what the formula (2.11) requires. The derivation of the result (2.11) assumed the spontaneous breaking of chiral symmetry and the existence of the Goldstone bosons. Thus, it refers only to the case Nf ≥ 2. For Nf = 1, there is no theoretical result, and no comparison can be done. It is amusing, however, that, if trying to ”continue analytically” the formula (2.11) down to the point Nf = 1, one would get a negative slope for ρ(λ) which agrees again with the instanton simulations. Qualitatively, the di?erent λ - dependence for di?erent Nf is natural. Mean spectral density ρ(λ) is obtained after averaging of the microscopic spectral density (1.3) over gluon ?elds with the weight which involves also the quark determinant [det(i D )]Nf ? ? 5
? ? λ2 n ?Nf

(2.12)

λn >0

The larger Nf is, the more the region of small λ is suppressed. There are good reasons to expect that, at Nf = 0, with no suppression at all, quark condensate ≡ ρ(0) is in?nite in the thermodynamic limit [8]. 3

3

Finite volume: the partition function.

In the rest of this talk, I derive and discuss some relations for the spectral density in the region of very small λ ? 1/| < q ?q > |V . These relations refer not to physical QCD in the in?nite volume, but to the theory de?ned in the ?nite Euclidean box. But, as our main goal is to discover islands of ?rm theoretical ground in the foreboding sea of numerical simulations , and the latter are done exclusively in ?nite volume, these intrinsically ?nite volume results can serve this purpose in exactly the same way as the result (2.11) derived in the in?nite volume. But before proceeding to the Dirac operator spectrum, we are in a position to study the more basic quantity — namely, the QCD partition function at ?nite volume. Speci?cally, we will be interested with the dependence of the partition function on the quark mass matrix M and the vacuum angle θ. Consider ?rst the theory with only one light quark ?avor. It involves a gap in the spectrum (U (1) axial symmetry is broken not spontaneously but explicitly by anomaly, and no goldstones appear). Thus, the extensive property for the partition function holds: Z ? exp{??vac (m, θ)V } (3.1)
1 If L ? Λ? QCD , the ?nite volume e?ects in ?vac are exponentially small [9]. Ward identities dictate that ?vac can depend on m and θ not in an arbitrary way, but only as a function of a particular combination meiθ . Expanding ?vac (meiθ ) in Taylor series (that makes sense as long as m ? ΛQCD ) and bearing in mind that ?vac is real, we get 2 Z ? eΣV m cos θ + O(m ) (3.2)

The parameter Σ is nothing else but the quark condensate (up to a sign). Really, <q ?q >0 = ? d ln Z = ?Σ dm (3.3)

Consider now the same problem but for several quark ?avors. Then spontaneous breaking of chiral symmetry occurs, goldstones appear, there is no gap in the spectrum
The instanton calculations of Ref.[6] do not exhibit the in?nite quark condensate for the quenched theory, but only a much more dramatic rise of the spectral density in the low λ region. But, probably, the ?nite value for ρ(0) is, again, a ?nite-volume e?ect.
3

6

, and the extensive property (3.1) does not hold anymore. More exactly, if quark masses are non-zero, the property (3.1) still holds when the length of the box L is much larger than the Compton wavelength of Goldstone particles ? 1/mφ ? 1/ mΛQCD . We will be interested, however, with the intermediate region
1 Λ? QCD ? L ?

1 mΛQCD

(3.4)

where the e?ects due to goldstones on the volume and mass dependence of the partition function are crucial. Fortunately, the goldstone properties at low energies (and, in the region L ? ?1 ΛQCD , only the low-energy properties of the Goldstone ?elds are relevant) are well known. They are described by the e?ective lagrangian (2.5) where, at nonzero θ , M should be substituted by Meiθ/Nf (again, the occurence of this particular combination is dictated by Ward identities). The partition function is given by the functional integral Z?
2 [dU ] exp ?Fπ V

d4 x

1 Tr{(?? U ? )(?? U )} + B Re Tr{Meiθ/Nf U ? 4
1 mφ

(3.5)

harmonics U0 of the ?eld U (x) is relevant, and the contribution of the higher harmonics in the integral (3.5) is suppressed. Then the functional integral is transformed into the ?nite-dimensional integral Z?
SU (Nf ) ? d?(U0 ) exp{V Σ Re Tr{Meiθ/Nf U0 }}

The crucial observation is that, as long as L ?

?√

1 , mΛQCD

only zero Fourier

(3.6)

The integral extends over the group SU (Nf ), and d?(U0 ) is the corresponding Haar measure. For Nf = 2, the integral (3.6) can be done explicitly. For Nf ≥ 3, it is not possible to do analytically in the general case. Sometimes (for the case M = m? 1), certain beautiful results can be obtained [2], but, to derive the sum rules for the small eigenvalues of the Dirac operator we are after in this talk, the closed analytic expression for Z (V, M, θ) is not actually required. The partition function (3.6) is a periodic function of θ with the period 2π . Thus, it can be presented as a Fourier series


Z (θ ) =
ν =?∞

Zν eiνθ

(3.7)

so that Zν = 1 2π
2π 0

dθe?iνθ Z (θ)dθ ?

U (Nf )

? ) (det U ? )ν exp{V Σ Re Tr{MU ? ? }} (3.8) d?(U

? = U0 e?iθ/Nf . with U 7

4

Finite volume: sum rules.

The main idea is basically the same as in Sect.2 — to compare the expression (3.8) derived in the chiral theory with the quark-gluon representation of the same quantity. Zν may be writen as a functional integral over the quark and gluon ?elds restricted on the topological class with a given winding number ν= 1 32π 2 ?a d 4 x Ga ?ν G?ν (4.1)

The integral over the Fermi ?elds produces the determinant of the Dirac operator, and we get for ν > 0 Zν = [dA]e? 4
1 a d4 xGa ?ν G?ν

(detf M)ν

λn >0

? detf (λ2 n + MM )

(4.2)

where the factor (detf M)ν arises due to the fermion zero modes λn = 0 which appear on a topologically non-trivial background due to the index theorem. (If ν is negative, the factor (detf M)ν is to be replaced by (detf M? )?ν ). Let us expand now Eqs.(3.8) and (4.2) in quark mass and compare the coe?cients of (detf M)ν Tr{MM? }. On the chiral side of the equality, we get some group integral which can be done explicitly. On the quark and gluon side, we get the expression involving < λn >0 1/λ2 n >ν where the average is de?ned as < f >ν = [dA]f e? 4 [dA]e? 4
1 1 a d4 xGa ?ν G?ν a d4 xGa ?ν G?ν

(

λn >0

λ2 n)

Nf

(

2 Nf λn >0 λn )

(4.3)

where the path integral is done over all gauge ?elds with the topological charge ν . Skipping the technical details, we present the ?nal result for the simplest sum rule thus derived: 1 V 2 Σ2 (4.4) = 2 4k λn >0 λn
ν

where k = |ν | + Nf . Expanding the expressions (3.8) and (4.2) further in quark mass and comparing the coe?cients of the two group invariants (detf M)ν (Tr{MM?})2 and (detf M)ν Tr{MM? MM? } , we get two di?erent sum rules for the inverse fourth powers of eigenvalues 1 4 λn >0 λn V 4 Σ4 ; = 16k (k 2 ? 1)
?

ν

1? ? 2 λn >0 λn

?2
ν

=

V 4 Σ4 16(k 2 ? 1)

(4.5)

(Two di?erent sum rules (4.5) are obtained when Nf ≥ 2. If Nf = 1, there is no di?erence between (Tr{MM? })2 and Tr{MM? MM? } and only one sum rule 8

involving inverse fourth powers of λn for the di?erence of the two sums entering Eq.(4.5) can be derived). Further expansion in M provides the sum rules with inverse sixth powers of eigenvalues etc. Note that the sums (4.4) , (4.5) are saturated by few ?rst eigenvalues. Therefore the sum rules provide the information on the very bottom of the ?nite-volume spectrum. The theoretical results (4.4) , (4.5) have not yet been confronted with lattice experiments. However, the numerical calculations in the instanton model in the case ν = 0 with di?erent number of quark ?avors are available. They are in a good agreement with the theory (see Fig.3 and the discussion thereafter).

5

Stochastic matrices

The sum rules (4.4) , (4.5) are exact theoretical results. They follow just from the form of QCD lagrangian and the assumption (which is the experimental fact in QCD) that the spontaneous breaking of chiral symmetry occurs. It is interesting, however, that much reacher information on the spectrum of Dirac operator in the region λ ? 1/(ΣV ) can be obtain if invoking a certain additional assumption on the properties of the functional integral in QCD [10]. Consider the Dirac operator in a particular gauge ?eld background with the topological charge ν . For simplicity, let us assume that the mass matrix is diagonal and real. Let us choose a particular basis E = {ψn (x)} and present the Dirac operator ?i D + mf as a matrix in this basis. The particular choice of the basis is not important for us, but a reader familiar with the model of instanton-antiinstanton liquid may think of ψn (x) as of zero-mode solutions for individual instantons and antiinstantons. What is important is that the basis E is chosen in such a way that all states ψn (x) have a de?nite (left or right) chirality. In that case, the number of left-handed states nL and the number of right-handed states nR in the basis should be related: nL ? nR = ν (5.1)

Thereby, the existence of exactly ν left-handed (ν > 0) or ?ν right-handed (ν < 0 ) zero eigenvalues of the massless Dirac operator ?i D is assured. In this basis, the full Dirac operator ?i D + mf can be presented as a matrix ? i D + mf mf iT ? = ? ? iT mf 9
? ?

(5.2)

where T is a general rectangular complex matrix nL × nR . We assume nL , nR to be very large and will eventually send them to in?nity. In this limit, the basis E spans the whole Hilbert space and the representation (5.2) is just exact. The partition function involves the Dirac determinant and the integral over all gluon con?gurations. If the basis E is ?xed (the same for all gluon ?eld con?gurations), the path integral over the latter can be traded for an integral over the matrices T . Thus, we have ? ? Nf m iT f ? Zν = D T P (T ) det ? ? (5.3) iT m f f

, the condition (5.1) for the dimensions of the matrix T being assumed. And now comes the crucial assumption. Let us assume that the measure P (T ) has the simple form A P (T ) = exp ? Tr{T T ? } + o(T T ? ) n (5.4)

The constant A will at the end determine the spectral density ρ(0) in the thermodynamic limit. The form (5.4) of the weight is rather natural. It respects chiral symmetry, analytic in T , and is conceptially similar to the Gibbs statistical distribution. However, a rigourous proof that P (T ) should have this form is absent by now. Therefore, the results for the microscopic spectral density ρ(λ ? 1/ΣV ) derived in ([10]) which follow from the representation (5.3) for the partition function and the heuristic assumption (5.4) have not the same status as the sum rules (4.4), (4.5) which are the theorems of QCD. Let us explain (very sketchy) how the results for the microscopic spectral density have been derived. For simplicity, assume that ν = 0 so that T is a square matrix: nL = nR ≡ n. P (T ) and the Dirac determinant depend only on the eigenvalues λn of the matrix T . Therefore, it is convenient to write the integral over D T as the integral over eigenvalues and do the integral over remaining angular variables on which the integrand does not depend. We have 4 DT = C
4

k<l

2 2 (λ 2 k ? λl )

λk dλk
k

(5.5)

This transformation and the whole stochastic matrix technique is well known for nuclear physicists and solid state physicists who are concerned with the problem of quantum dynamic for stochastic hamiltonia. The particular problem studied here coincides with the known problem of ”Gaussian unitary ensemble”. The vanishing of the Jacobian at λk = λl re?ects the notorious repulsion of the levels. For an extensive review of the relevant mathematics see [11].

10

Thus, joint eigenvalue distribution is just ρ(λ1 , . . . , λn ) ?
2 2 (λ 2 k ? λl )

k<l

k

λk exp{?

A n 2 λ } n k=1 k

2 (λ 2 k + mf ) f

(5.6)

The microscopic spectral density is obtained after integration of (5.6) over all eigenvalues but one. After some work, one gets the answer in the limit n → ∞. In [10] the result was obtained in the particular case ν = 0. Being expressed via physical variables V (the volume of the system) and Σ (the quark condensate), A ≡ (ΣV )2 /4, it has the form ρ(λ) = Σ2 V λ 2 JNf (ΣV λ) ? JNf +1 (ΣV λ)JNf ?1 (ΣV λ) 2 (5.7)

where J? (x) are the Bessel functions. The result (5.7) has been obtained in [10] under the assumption ν = 0. In [12] it has been generalized to the case ν = 0. [the only change is that Nf should be substituted by Nf + |ν | as in Eqs. (4.4, 4.5)]. At very small λ ? 1/V Σ, the spectral density is suppressed ρ(λ) ? λ2Nf +1 (the suppression is due to the determinant factor (2.12) which punishes small eigenvalues). But then it rises and after several oscillations with decreasing amplitude levels o? at a constant ρ(0) = Σ/π as it should in the thermodynamic limit V → ∞. The function (5.7) with Nf = 0, 1, 2 together with the results of numerical instanton calculations are plotted in Fig.3 borrowed from Ref.[6]. Likewise, one can integrate Eq.(5.6) over all eigenvalues but two and derive the expression for the correlation function ρ(λ1 , λ2 ) etc. The sum rules derived in the previous section can be expressed as the integrals of ρ(λ), ρ(λ1 , λ2 ) etc. with a proper weight. For example, the simplest sum rule is 1 2 λn >0 λn


=
ν =0

0

ρ(λ)dλ V 2 Σ2 = λ2 4Nf

(5.8)

which coincides with (4.4). Thus, the excellent agreement of the model instanton calculations with the stochastic matrix model results displayed in Fig.3 implies also the agreement of the former with the exact theoretical results (4.4), (4.5) etc.

6

Exotic theories.

The same program as in QCD can be carried out in other gauge theories with nonstandard fermion content. The results derived earlier depended on the assumption of the standard pattern of chiral symmetry breaking SUL (Nf ) ? SUR (Nf ) → SUV (Nf ) 11 (6.1)

2 so that Nf ? 1 Goldstone bosons living on the coset, which is also SU (Nf ), appear. However, the breaking according to this scheme only occurs for fermions belonging to the complex (e.g. fundamental) representation of the gauge group. The SU (Nc ) group involves truly complex representations when Nc ≥ 3. For Nc = 2, the fundamental representation (and also other representations with half-integer isospin) is pseudoreal: quarks and antiquarks transform in the same way under the action of the gauge group and the pattern of chiral symmetry breaking is di?erent leading to di?erent sum rules. A third pattern of chiral symmetry breaking is for fermions in the real (e.g. adjoint) representation leading to yet another class of sum rules. Consider ?rst the case of pseudoreal fermions. It may have a considerable practical importance as the numerical calculations with the SU (2) gauge group are simpler than with SU (3) group, and it may be easier to confront the sum rules with lattice simulations. The true chiral symmetry group of the QCD-like lagrangian but with SU (2) gauge group is SU (2Nf ) rather than just SUL (Nf ) ? SU (NR ) ? UV (1) (it involves also the transformations that mix quarks with antiquarks). The pattern of spontaneous symmetry breaking due to formation of quark condensate is [13]

SU (2Nf ) → Sp(2Nf ) The breaking (6.2) leads to the appearance of
2 2 (2Nf )2 ? 1 ? (2Nf + Nf ) = 2Nf ? Nf ? 1

(6.2)

Goldstone bosons which are parametrized by the coset SU (2Nf )/Sp(2Nf ). For Nf = 2 , we have 5 instead of the usual 3 Goldstone bosons. Let us now allow for small non-zero quark masses (which break the chiral symmetry explicitly and give non-zero masses to Goldstone particles) and study the dependence of the partition function in the sector with a given topological charge ν on the quark mass matrix M and the volume V . The analog of the result (3.8) for the pseudoreal case which is valid in the region (3.4) is the following (see [14]) for more details) Zν ? where d?(U )(detU )?ν exp VΣ ReTr{MUIU T } 2
?

U (2Nf )

(6.3)

is the symplectic 2Nf ? 2Nf matrix. Formally, the representation (6.3) involves the integral over U (2Nf ), but actually it is the integral over the coset SU (2Nf )/Sp(2Nf ) involving less number of parameters (and the integral over the U (1) part of U (2Nf ) 12

0 ? 1 ? I=? ? ?1 0

?

(6.4)

which is nothing else but a vacuum angle θ) — multiplying U by a simplectic matrix ∈ Sp(2Nf ) leaves the integrand invariant. Expanding (6.3) over the group invariants involving the quark matrix M and expanding also the quark-gluon representation of the same partition function, the sum rules for the inverse powers of the eigenvalues can be derived. The simplest sum rule is V 2 Σ2 1 (6.5) = 2 4(|ν | + 2Nf ? 1) λn >0 λn
ν

Consider now the third non-trivial class of theories where the fermions belong to the real representation of the gauge group. The supersymmetric Yang-Mills theories involving Majorana fermions in the adjoint color representation belong to this class. We will not assume, however, that the theory is supersymmetric and consider a YangMills theory coupled to Nf di?erent Majorana adjoint fermion ?elds. Also, we will assume that the fermion condensate is formed and the chiral symmetry which the lagrangian enjoys for Nf ≥ 2 is spontaneously broken 5 . The chiral symmetry group of such a theory is SU (Nf ). Formation of the condensate breaks it down to SO (Nf ), and
2 Nf ?1?

Nf (Nf + 1) Nf (Nf ? 1) = ?1 2 2

Goldstone bosons appear. Again, the partition function Zν is given by the integral over the coset SU (Nf )/SO (Nf ) which has the form Zν ?
U (Nf )

d?(U )(detU )?2νNc exp V Σ Re Tr{MUU T }

(6.6)

The essential di?erence with the case of fundamental fermions is that the admissible values for the topological charge ν are not integer but integer multiples of 1/Nc . That means, in particular, that the function Z (M, θ) given by the Fourier sum (3.7) is a periodic function of θ with the period 2πNc (not 2π as before). See Ref.[2] for a detailed discussion of this important issue. To derive the sum rules, we have to expand (6.6) over M and to compare it with the expansion of the quark-gluon path integral for the partition function. And here we meet a known problem characteristic for theories with Majorana particles. The matter is that the reality condition for the Fermi ?elds can be ful?lled in Minkowski space but not in Euclidean space - it immediately leads to a contradiction [16]
This dynamic assumption is not so innocent. For example, in N = 2 supersymmetric YangMills theory it does not happen [15]. But this theory involves also massless adjoint scalar ?elds with particular interaction vertices. It is quite conceivable that the non-supersymmetric theory with only gauge ?elds and two Majorana adjoint ?avors involves chiral symmetry breaking. All the results are derived under this assumption.
5

13

However, the Euclidean path integral for the partition function in a theory with Majorana fermions still can be de?ned by analytic continuation of the Minkowski space path integral. In the Minkowski space, the path integral over fermion ?elds is equal to the square root of the corresponding Dirac determinant. The latter can be continued to Euclidean space without problems, and the square root is also taken easily here due to a notable fact that the spectrum of the Euclidean Dirac operator for the adjoint fermions is doubly degenerate: for any eigenfunction un (x) with an eigenvalue λn , the function C ?1 u? n (x) (C is the charge conjugation matrix) is linearly independent from un (x) and has exactly the same eigenvalue [2]. Thus, square root of the Dirac determinant is just ? ]1/2 = (detf M)νNc [det(?i D + M
′ λn >0 ? detf (λ2 n + MM )

(6.7)

? = 1 (1 ? γ 5 )M + 1 (1 + γ 5 )M? , the product counts only one eigenvalue of where M 2 2 each degenerate pair, and the factor (detf M)νNc re?ects the presence of νNc pairs of zero eigenmodes of adjoint Dirac operator in the gauge ?eld background with the topological charge ν 6 . The ?nal form of the simplest sum rule for the eigenvalues of the Dirac adjoint determinant is ′ V 2 Σ2 1 (6.8) = 2 4(|ν |Nc + (Nf + 1)/2) λn >0 λn
ν

where, again, only one eigenvalue of each degenerate pair is taken into account in the sum. The sum rules (4.4), (6.5), and (6.8) can be written universally as 1 2 λn >0 λn =
ν

V 2 Σ2 4{|ν | + [dim (coset) + 1]/Nf }

(6.9)

with the rescaling ν → νNc and counting in the sum only one eigenvalue of each degenerate pair in the adjoint case. The sum rules (6.5), and (6.8) and their analogs with higher inverse powers of λn can be derived also in the stochastic matrix technique, and not only the sum rules, but also the expressions for the microscopic spectral density ρ(λ), correlators ρ(λ1 , λ2 ) etc. [18] Mathematically, 3 classes of theories discussed (with complex fermions, pseudoreal fermions and real fermions) are described in terms of 3 classic stochastic enThe nice result (6.7) is speci?c for the theories with adjoint Majorana fermions. To de?ne in the Euclidean space a theory like the standard model involving chiral fermions in the fundamental representation is more di?cult [17].
6

14

sembles: Gaussian unitary, Gaussian orthogonal, and Gaussian simplectic. It is noteworthy that, though the sum rules in all 3 cases are the same for Nf = 1 [where no spontaneous breaking of chiral symmetry occurs, no Goldstone bosons appear, and the partition function has the extensive form (3.1)] , the microscopic spectral densities and the correlators are not. That elucidates again the fact that , when deriving microscopic spectral densities in the stochastic matrix technique, an extra assumption (5.4) beyond using just the symmetry properties of the theory is neccessary to adopt.

7

Acknowledgements.

It is a pleasure for me to thank the organizers of the Seoul meeting for kind hospitality. I am indebted to J. Verbaarschot for valuable discussions and for kindly sending me the postscript ?les of Fig.2 and Fig.3. This work was supported in part by Schweizerisher Nationalfonds and the INTAS grant 93-0283.

References
[1] T.Banks and A.Casher, Nucl. Phys. B168 (1980) 103. [2] H. Leutwyler and A.V. Smilga, Phys. Rev. D46 (1992) 5607. [3] A.V. Smilga and J. Stern, Phys. Lett. B318 (1993) 531. [4] See e.g. H. Leutwyler, Chiral E?ective Lagrangians, in: Perspectives in the Standard Model. Proc. 1991 Theor. Adv. Study Institute in Elementary Particle Physics (World Scienti?c, 1992). [5] N. Christ, Talk given at ICHEP conference in Glasgow, July 1994, Unpublished. [6] J. Verbaarschot, Acta Phys. Polonica 25 (1994) 133. [7] N. Christ, Private communication. [8] A.V. Smilga, Phys. Rev. D46 (1992) 5598. [9] C.N. Yang and T.D. Lee, Phys. Rev. 87 (1952) 404. [10] J. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852. [11] M.Mehta, Random matrices, Academic Press, San Diego, 1991. [12] J.Verbaarschot, Nucl.Phys. B426 (1994) 559. 15

[13] S. Dimopoulos, Nucl. Phys. B168 (1980) 69; M. Peskin, Nucl. Phys. B175 (1980) 197; M. Vysotsky, I. Kogan, and M. Shifman, Sov. J. Nucl. Phys. 42 (1985) 318. [14] A.V. Smilga and J. Verbaarschot, Phys. Rev. D51 (1995) 829. [15] N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19 and references therein. [16] P. Ramond, Field theory: A Modern Primer, Benjamin/Cunnings, Reading, MA, 1981. [17] A.I. Vainshtein and V.I. Zakharov, JETP Lett. 35 (1982) 323. [18] J. Verbaarschot, Phys. Rev. Lett. 72 (1994) 3531.

Figure captions.
Fig. 1. 2-point correlator. (a) Quark representation. The loop is evaluated in a gauge-?eld background to be averaged over afterwards. (b) Quasi-Goldstone contribution. Fig. 2. Instanton calculations for the spectral density n(λ) = ρ(λ) for di?erent Nf . λ is measured in units of ΛQCD , and the area below the curve is normalized to 1. Fig. 3. Microscopic spectral density as a function of z = ΣV λ/π . The normalization ρ(0) = 1 is chosen. Dashed lines correspond to the theoretical result (5.7). Full lines present the results of numerical calculations in the instanton liquid model.

16

background

a)

b)


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