aa r X i v : . [ h e p - t h ] F e b Proper-time method for unequal masses
A. A. Osipov
Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, 141980 Dubna, Russia
The result of removing of heavy non-equal mass particles from the theory can be described, at lowenergy, by the effective action, which is an asymptotic series in inverse-square powers of the mass.Here it is shown how the leading terms of the series can be calculated based on the Fock-Schwingerproper-time method. To illustrate the idea, the theory with broken U (3) × U (3) chiral symmetryis considered. We demonstrate that unequal masses give rise to a set of new vertices describing theconsequences of explicit flavour symmetry breaking, which do not arise in a less rigorous approach.The result can be used in the study of theories with explicit and spontaneous chiral symmetryviolation, or within the context of the standard and beyond standard model effective field theory. Quite often it becomes necessary to calculate the de-terminant of a positive definite elliptic operator that de-scribes quadratic fluctuations of quantum fields in pres-ence of some external fields. One of the results of suchcalculations can be an asymptotic expansion of the in-duced effective action in powers of proper-time [1–3] withSeeley-DeWitt coefficients a n ( x, y ) [4–6]. These coeffi-cients are polynomials in the classical fields and describe,in the coincidence limit y → x , the local vertices of thecorresponding effective Lagrangian.When quantum fields have large equal masses m , it isconvenient to resort to an expansion in powers of proper-time to obtain an expansion in the inverse powers of mass,which is valid when all classical fields and their deriva-tives are small compared to the mass of the quantumfield. In this case, the asymptotic coefficients a n do notchange. Such long wavelengths ( λ ≫ /m ) expansionallows to obtain an action that takes into account ef-fectively the leading low energy effect of virtual heavystates. This scenario is realized in theories with spon-taneously broken symmetry, or in the theories with alarge bare mass. A typical example of the first type isthe Nambu – Jona-Lasinio (NJL) model [7, 8], where theground state in the strong coupling regime is found tobe separated by a gap from the excited states (nucle-ons). Reinterpreting nucleons as quarks, one obtains alow energy meson action from one-loop quark dynam-ics [9]. The proper-time method is especially useful here[10, 11]. Examples of the second type arise under ex-tension of some effective field theory X with symmetrygroup G to another effective theory X ′ , when X ′ containsheavy degrees of freedom belonging to some representa-tion of G . At present, such theories are being activelystudied in the context of extending the standard modelof electroweak interactions [12].In realistic models one is often confronted with thedifficulty that the flavor symmetry is broken by largeunequal masses M = diag( m , m , . . . , m f ). In this casethe complete factorization of M in the heat kernel isimpossible because M does not commute with the restof an elliptic operator. As a result, the asymptotic serieshas a much more complicated structure [13–15]. In thisnote, we propose a new solution to this problem, which,we hope, can be useful in practical applications. For definiteness, we consider a theory with U (3) × U (3)chiral symmetry broken down explicitly and sponta-neously to its subgroups. It is well-known that the group U (3) L × U (3) R corresponds to the symmetry of the QCDLagrangian in the ideal world of massless quarks in thelarge N c limit. In the real world with nonzero bare massesof current quarks, chiral symmetry is explicitly broken.At low energies, as a result of dynamic breaking of chiralsymmetry, a gap appears in the spectrum of fermions,quarks become much more heavier and acquire unequalmasses. The latter makes it possible to construct an ef-fective low-energy meson Lagrangian in the form of 1 /M series. The method described below, in particular, canbe used for obtaining such effective Lagrangian in theframework of the NJL model, as an alternative to theapproaches developed in [11, 16].We shall be mainly concerned here with the asymp-totic behavior of the following object in Euclidean four-dimensional space W E = − ln | det D E | = 12 Z ∞ dtt ρ Λ ( t ) Tr (cid:16) e − tD † E D E (cid:17) . (1)It represents the real part of the one-quark-loop contribu-tion to the meson effective action in form of the proper-time integral. The integral diverges at the lower limit,therefore, a regulator ρ Λ ( t ) is introduced, where Λ is acutoff. The Dirac operator in Euclidean space, D E , hasthe form D E = iγ α d α − M + s + iγ p, (2)where d α = ∂ α + i Γ α , Γ α = v α + γ a α , α = 1 , , ,
4. Theexternal scalar s , pseudoscalar p , vector v α and axial-vector a α fields are embedded in the flavor space throughthe set of matrices λ a = ( λ , λ i ), where λ = p / λ i are the eight SU (3) Gell-Mann matrices; for instance, s = s a λ a , and so on for all fields. The quark massesare given by the diagonal matrix M = diag( m , m , m )in the flavor space. The symbol ”Tr” denotes the traceover Dirac ( D ) γ -matrices, colour ( c ) SU (3) matrices,and flavor ( f ) matrices, as well as integration over co-ordinates of the Euclidean space: Tr ≡ tr I R d x , where I = ( D, c, f ). The trace in the color space is trivial: itleads to the overall factor N c = 3. The dependence onexternal fields in D E after switching to the Hermitianoperator D † E D E = M − d α + Y (3)is collected in Y and the covariant derivative d α . In thefollowing we do not need the explicit form of Y .To advance in the evaluation of expression (1), we usethe Schwinger technique of a fictitious Hilbert space [6].It allows the effective action to be represented as an in-tegral over the 4-momenta k α W E = 12 Z d x Z d k (2 π ) e − k × ∞ Z dtt ρ Λ ( t ) tr I (cid:16) e − t ( M + A ) (cid:17) , (4)where A = − d α − ik∂/ √ t + Y . Since A contains openderivatives with respect to coordinates, it is still neces-sary to clarify the meaning of tr I . The space x ∈ R hasno boundaries, therefore we can integrate by parts. How-ever, this operation will be unambiguous only if tr I doesnot change when its x -dependent elements are cyclicallyrearranged. This property can be ensured by the explicitsymmetrization, i.e., tr I in such cases is understood asstr I ( A A . . . A n ) = 1 n X cycl.perm. tr I ( A A . . . A n ) . (5)To distinguish the leading contributions in the expan-sion in 1 /M , it suffices to take into account terms of atmost t order, which under the flavor trace have the formtr f (cid:16) e − t ( M + A ) (cid:17) = X i =1 c i ( t ) − t X i =1 c i ( t ) tr f A i + t X i,j c ij ( t ) tr f ( A i A j ) − t X i,j,k c ijk ( t ) tr f ( A i A j A k )+ t X i,j,k,l c ijkl ( t ) tr f ( A i A j A k A l ) + O ( t ) . (6)Here A i = E i A , where 3 × E i ) nm = δ in δ im ;the coefficients c i i ...i n ( t ) are completely symmetric withrespect to any permutation of indices and are well known[17–19]. They depend on quark masses and proper-time.In the case of equal masses, all coefficients coincide, thatis, c i ( t ) = c ii ( t ) = c iii ( t ) = c iiii ( t ) = e − tm i . It also fol-lows from the formula (6) that the number of indices inthe coefficient c i i ...i n is equal to the number of oper-ators A i , representing external fields. Hence, there is arelation between the number of indices in c i i ...i n andthe contribution of the one-loop diagram with n externallegs. The formula (12) will clarify this correspondence.Substituting (6) into (4) and integrating over momenta k α and proper-time t , we arrive at the expression W E = N c π Z d x ∞ X n =0 tr Df b n ( x, x ) , (7) where coefficients b n ( x, x ) depend on the external fieldsand quark masses, i.e., they contain information aboutboth the effective meson vertices and corresponding cou-pling constants. If all masses are equal, the dependenceon m = m i is factorized in form of the integral [18] J n ( m i ) = ∞ Z dtt − n ρ Λ ( t ) c i ( t ) (8)and the field-dependent part takes a standard Seeley-DeWitt form a n ( x, x ). For large masses m , the coeffi-cients b n ( x, x ) exhibit the same asymptotic behavior as J n − ( m ), i.e., b n ∼ m − n − .Consider the leading terms b and b . The case n = 0is of no interest because b contains no fields and can beomitted from the effective action. The coefficients with n ≥ b and b .For convenience of writing the result of our calcula-tions, along with the usual matrix multiplication, we willuse the non-standard Hadamard product [20], which isthe matrix of elementwise products( A ◦ B ) ij = A ij B ij . (9)The Hadamard product is commutative unlike regularmatrix multiplication, but the distributive and associa-tive properties are retained. It has previously beenproven to be an useful tool when non-degenerate massmatrices and non trivial flavor symmetry contractionswere involved [21].Now, the heat coefficient b ( x, x ) can be written as b = − J ◦ Y, (10)where ( J ) ij = δ ij J ( m i ) is a diagonal matrix with el-ements given by (8) (for n = 0). This matrix containscontributions of the Feynman one-loop diagram, knownas a ”tadpole”. The regularization should be chosen inaccord with the problem studied, for the NJL model thiscan be ρ Λ ( t ) = 1 − (1 + t Λ ) e − t Λ .Consider the second coefficient b ( x, x ). After sometiring calculations we can represent the result in the form b = 12 Y ( J ◦ Y ) −
112 Γ αβ (cid:0) J ◦ Γ αβ (cid:1) + ∆ b , (11)where the antisymmetric tensor Γ αβ = F αβ + i [Γ α , Γ β ], F αβ = ∂ α Γ β − ∂ β Γ α , and J ij = J ( m i , m j ) is a symmetric3 × → ∞ ) of Feynman self-energy diagramswith masses of virtual particles m i and m j . Since wealso need expressions for the similar contributions comingfrom triangular J ijk and box J iijk diagrams, we collectthem together in one formula J ij J ijk J iijk = ∞ Z dtt ρ Λ ( t ) c ij ( t ) c ijk ( t ) c iijk ( t ) . (12)In the theory under consideration, there are only threeflavors of quarks, therefore in J iijk at least two indicescoincide. From the known properties of c i i ...i n ( t ) onecan deduce that J ii = J iii = J iiii . The integrals take thisform when the quark masses are equal. Their coincidenceis an important property which is used to demonstratethat ∆ b vanishes in the equal-mass limit.In the limit of equal masses, the first two terms in (11)yield the well-known result [5], and the third one van-ishes, that is, it contains only contributions associatedwith an explicit violation of chiral symmetry. Withoutthis term, the description of exact breaking of flavor sym-metry is incomplete. To write an expression for ∆ b , letus consider the different contributions in∆ b = Ω Y + X n =0 Ω ( n ) , (13)where Ω Y is the sum of all terms linear in Y . Ω ( n ) is thesum of terms with n derivatives which consists of onlyspin-one fields.For Ω Y we haveΩ Y = AY + i R ◦ Γ α ) ↔ ∂ α Y, (14)where R and A are 3 × R ij = −
12 ( J iij − J jji ) , (15) A ii = X j = i ( J ii − J ij + R ij ) Γ αij Γ αji ,A ij = (cid:2) ( J ij − J ijk )Γ αik Γ αkj − R ij (cid:0) Γ αij Γ αjj − Γ αii Γ αij (cid:1)(cid:3) | i = j = k , (16)and the left-right derivative is defined by the difference Y ↔ ∂ α Γ α = Y ∂ α Γ α − ( ∂ α Y )Γ α . To avoid misunderstand-ings, we emphasize that repeated flavor indices do not im-ply summation over them. Here and below, summationis carried out only if the summation symbol is explicitlywritten.Let us consider the terms with two derivativesΩ (2) = 124 (cid:2) F αβ (cid:0) T ◦ F αβ (cid:1) + 6( ∂ Γ) ( T ◦ ∂ Γ) (cid:3) . (17) The first term makes an additional contribution to thekinetic part of the effective Lagrangian of spin-1 fieldsdescribed by the second term in (11). The symmetricmatrix T is defined by T ij = J ij − J iijj . The shorthand ∂ Γ ≡ ∂ α Γ α implies summation over omitted indices α .In applications, one can omit the second term in (17)to ensure that the energy of the massive spin-1 field ispositive definite.The terms with one derivative can be collected in theexpressionΩ (1) = i C αβ F αβ + i δ αβγσ ( K ◦ ∂ α Γ β ) E γσ + 3 i (cid:0) T ◦ ∂ α Γ β (cid:1) L βα − , (18)which represents the effective three-particle verticesdescribing the local interaction of vector and axial-vector fields. Symbol δ αβγσ = δ αβ δ γσ + δ αγ δ βσ + δ ασ δ βγ is a totally symmetric tensor composed of the product oftwo Kronecker symbols. The diagonal and off-diagonalelements of the matrix C αβ are, respectively, of the form C αβii = X j = i (cid:18) J ii − J ij + 14 R ij + 34 T ij (cid:19) Γ αij Γ βji ,C αβij = ( J ij − J )Γ αik Γ βkj | i = j = k + 14 R ij (cid:16) L αβ + (cid:17) ij + 34 T ij (cid:16) L αβ − (cid:17) ij , (19)where we use the notation (cid:16) L αβ ± (cid:17) ij = (cid:0) Γ αii ± Γ αjj (cid:1) Γ βij . (20)Note that elements of the matrix Γ αij commute with eachother. The antisymmetric matrix K is defined by theexpression K ij = ( J jjik − J iijk ) | i = j = k . (21)The tensor E γσ has no diagonal elements too, althoughit is not antisymmetric E γσii = 0 , E γσij = Γ γik Γ σkj , ( i = j = k ) . (22)The last thing left to consider is the term withoutderivatives Ω (0) , which is a sum of the terms ∝ Γ . Sincethere are many of those, it is convenient to present theresult in a form where the trace over flavor indices isalready calculated. Then we havetr f Ω (0) = X i 23 ( J ii + J jj ) − J ij (cid:21) (Γ ij Γ ji ) + 16 (2 J ij − J ii − J jj ) (Γ ij Γ ij ) (Γ ji Γ ji ) (cid:27) + X i = j = k,j 13 ( J jk + 2 J iijk ) + J ii − J ij − J ik (cid:21) (Γ ij Γ ji ) (Γ ik Γ ki ) + (cid:20) 13 ( J ii + 2 J iijk ) + J jk − J ijk (cid:21) (Γ ji Γ ik ) (Γ ki Γ ij )+ 13 (2 J iijk − J ij − J ik ) [(Γ ii Γ jk ) (Γ ij Γ ki ) + (Γ ik Γ ji ) (Γ kj Γ ii )] + 13 (2 J iijk − J ii − J jk ) (Γ ij Γ ik ) (Γ ki Γ ji ) (cid:27) + X i = j = k (cid:26) 13 ( J ik + 2 J iijk − J ijk ) [(Γ ii Γ ij ) (Γ jk Γ ki ) + (Γ ik Γ kj ) (Γ ji Γ ii )] + R ij h(cid:0) L αα − (cid:1) ij (Γ jk Γ ki ) + (Γ ij Γ ji ) (Γ ik Γ ki ) i(cid:27) + 13 δ αβγσ X i = j T ij (cid:18) Γ αii Γ βii Γ γij Γ σji − Γ αii Γ βij Γ γjj Γ σji − 12 Γ αij Γ βji Γ γij Γ σji (cid:19) . (23)Here, the expression in the parentheses (Γ ij Γ ji ) is un-derstood to be summed over alpha Γ αij Γ αji . In the limitof equal quark masses (23) is zero, since all J -dependentfactors vanish.Let’s summarize. Above, we presented a new methodfor obtaining the effective Lagrangian in a theory withheavy particles of unequal masses and a non-trivial fla-vor symmetry. Unlike [13], where authors use the mod-ified DeWitt WKB form and solve recurrent equationsto determine the heat coefficients, our calculations arebased on the formula (6), which allows us to resum theproper-time series in accordance with the contributionsof corresponding one-loop Feynman diagrams with therequired number of external fields. As a result, we arrive at an effective action (7), in which each of the coefficientscan be calculated independently of the other, and corre-sponds to a certain order of 1 /M expansion. We haveexplicitly calculated two leading coefficients for the caseof broken U (3) × U (3) symmetry. 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