arXiv:hep-th/9802043v5 17 Oct 1998 ofra nml;3[etr h iao hne h conforma the changes dilaton The 3)[Vector] anomaly; conformal o ahcs:1[clr h iao hne h ofra anom invariant, conformal the conformal charact changes anomaly new dilaton conformal The a the 1)[Scalar] pla affects case: field dilaton each the dilaton for anomaly that the conformal shown the of is of shift structure It general constant dilato The the the role. of involving important symmetry terms, The trivial and d in obtained. invariants is anomalies conformal analysis a conformal cohomology new fields explicit The some vector their found. and and are background spinor fields gravity-dilaton quantum t Scalar, matter of measure. the hermiticity as of derivatives, choice tak total is and of care operator treatment Special fields, studied. of systematically is rescaling dilaton, the are which with theories, coupled gravity-matter 4D for anomaly conformal The Abstract ofra nml n4 Gravity-Matter 4D in Anomaly Conformal 2 1 † -al:[email protected] : e-mail [email protected] : e-mail os eaoia nvriy 301Tmk RUSSIA Tomsk, 634041 University, Pedagogical Tomsk hoisNnmnmlyCuldwith Coupled Non-minimally Theories hih ICHINOSE Shoichi eateto hsc,Uiest fShizuoka of Universuty Physics, of Department aa5-,Siuk 2-56 JAPAN 422-8526, Shizuoka 52-1, Yada I 4 )Sio]Tedltndoes dilaton The 2)[Spinor] ; 1 Dilaton n egiD ODINTSOV D. Sergei and 1 hep-th/9802043 not sexamined. is non-minimally † l nyby only aly hnethe change esystem he anomaly l eristically 2 US-98-01 etaken re n and one nfor: en ,are n, san ys the by three new (generalized) conformal invariants, I4,I2,I1. We present some new anomaly formulae which are useful for practical calculations. Finally, the anomaly induced action is calculated for the dilatonic Wess-Zumino model. We point out that the coefficient of the total derivative term in the conformal anomaly for the 2D scalar coupled to a dilaton is ambiguous. This resolves the disagreement between calculations in refs.[3, 4, 5, 6] and the result of Hawking-Bousso[2].
PACS NO: 04.62.+v,11.25.Db,11.30.-j,14.80.-j • Keywords: conformal anomaly, dilaton gravity, dilatonic Wess-Zumino • model, anomaly formula
1 Introduction
Stimulated by string theory, the higher dimensional approach to the 4D the- ory becomes gradually important. Indeed it provides fruitful possibilities for theoretical analysis. Kaluza-Klein type dimensional reduction is the most popular one. However realizations of such reductions depend on the chosen higher dimension, the method of reduction, the choice of a symmetry ansatz, and so on. At present there seems to be no fixed rule to find the ”right” one. We first note that, in low-energy effective actions, some characteristic fields, such as the dilaton and the anti-symmetric tensor commonly appear as rem- nants of the extra dimensions. Through the study of the dynamical effect of such fields, we can learn some common features of 4D effective theories that the higher-dimensional theories produce in the low energy limit. We focus on the dilaton field for simplicity. Its effect on the Weyl anomaly is examined. We start from the Weyl invariant action in order to set up the problem clearly. The coupling of the dilaton is fixed by keeping the Weyl symmetry. The most characteristic aspect is its exponential form. Further- more it couples to most fields just like the gravitational field itself. ( From
2 the standpoint of the string theory or Kaluza-Klein theory, this is a natural aspect because the dilaton comes from a gravitational component in the com- pactified dimensions.) We treat the dilaton as the background field in the same way as the gravitational field. We take the standpoint that the quan- tum treatment of the dilaton, like the gravitational field, should be explained by the string theory. It is interesting that the dilaton is expected to play a role of a coupling in string theory. Indeed, in the present treatment, the dilaton is invariant under all symmetry transformations and behaves just like a coupling constant except that it depends on the space coordinates. The dilaton transforms as a scalar under the general coordinate transformation. Under the Weyl transformation, it does not change
φ′ = φ . (1)
In this point the dilaton transforms differently from the scalar field except in the 2D space (see Appendix C), which is natural because of the dimensional difference between the dilaton and a scalar field. As for the role of the dilaton, some interesting arguments have been presented recently. Even at the classical level, the dilaton can make the singularity behaviour weaker in the early universe cosmology[1]. In 2D or 4D black hole physics, its presence influences the Hawking radiation [2, 3, 4, 5, 6]. As for a recent development using the conformal anomaly, we note ref.[7] where a nonperturbative approach to the conformal field theory in higher dimensions is presented ( for early works in the study of higher dimensional conformal field theory, see[8]). The conformal anomaly in 4D dilaton gravity- vector has been recently found in [9] where the induced action is examined and by [10] where the general structure of the conformal anomaly is analysed. The above is the general background to the present work. In the following we will concretely analyse the following points.
1. We take the spin=0,1/2,1 matter fields on dilatonic curved backgrounds and examine the conformal anomaly due to quantum effects. The dila- ton’s presence is a new point here.
2. We are interested in the general structure of the conformal anomaly.
3 3. In the evaluation of the anomaly we pay attention to the following points: the arbitrariness of total derivatives or partial derivatives in the operators, rescaling of fields and the choice of measure.
4. We present some useful anomaly formulae and show how to use them.
5. We check the previous work, which is obtained as a special case : dilaton field = const by using a different approach (heat-kernel regularization and Fujikawa method).
We begin with the explanation of the present formalism and the coho- mology analysis in Sec.2. In Sec.3 the scalar theory is considered. This is the simplest case and is the best model in which the present analysis is explained. In Sec.4 the vector theory is considered. We must take into account the gauge fixing which introduces some complication. It breaks the Weyl symmetry as well as the gauge symmetry. The final result is therefore highly non-trivial. In Sec.5 we treat the spinor theory. It turns out that the dilaton does not affect the spinor theory. Finally we discuss the anomaly induced effective ac- tion in Sec.6. As an example the dilatonic Wess-Zumino model is considered. We conclude in Sec.7. In appendix A and B, we derive useful anomaly for- mulae used in the text. We use heat-kernel regularization for the ultra-violet divergences and follow the Fujikawa method[11]. The results turn out to be the same as the counterterm formulae already obtained by other authors, where dimensional regularization was used and the (log-)divergent part of the effective action was calculated. In the present approach the anomaly term is definitely determined (no ambiguity of the total divergence) once the system operator (elliptic differential operator) and the regularization are fixed. In appendix C the Weyl transformations of various fields are listed. In appendix D, the list of ”trivial terms” is presented. In appendix E we discuss the conformal anomaly for a two-dimensional dilaton coupled scalar.
2 Formalism and Cohomology Analysis
We consider the system of quantum matter (scalar,vector,spinor,etc.) fields f(x), in the background metric gµν(x) and the background dilaton field φ(x), described by the action S[f; g,φ]. The (Wilsonian) effective action Γ[g,φ] is
4 defined by
eΓ[g,φ] = f exp S[f; g,φ] . (2) Z D { } Under the Weyl transformation, the classical lagrangian is assumed to be invariant.
2α(x) cα(x) φ′ = φ , gµν′ = e gµν , f ′ = e f , S[f ′; g′,φ′ = φ]= S[f; g,φ] , (3) where c is some constant (the scale dimension of the matter field). The effective action Γ is generally changed due to the measure (quantum) effect which is the origin of the conformal anomaly[11]. The change is given by
δΓ δΓ µν =2gµν(x) 2gµν < T > , (4) δα(x) δg (x) ≡ α=0 µν
where T µν is the energy-momentum tensor. This quantity is the conformal (Weyl) anomaly and is evaluated as follows.
Γ[g′,φ′=φ] e = f ′ exp S[f ′; g′,φ′ = φ] Z D { } ∂f = f det( ′ )exp S[f; g,φ] . (5) Z D ∂f { } We can evaluate the Jacobian in the heat-kernel regularization.
∂f ln det ′ = Tr ln ecα(x)δ4(x y) = c Tr α(x)δ4(x y) + O(α2) ∂f { − } { − } tD~ 2 = c lim Tr α(x) < x e− y > + O(α ) , t +0 | | → n o µν c tD~ g < T >= tr < x e− x> 0 . (6) µν 2 | | |t n o By differentiating the equation (5) w.r.t.(with respect to) the background sources gµν and φ, we can obtain various Ward-Takahashi identites which take into account the conformal anomaly. They correspond to 1-loop graphs of 2-point, 3-point, ... functions of corresponding currents. Taking t0 part in (6) corresponds to considering the log-divergent part in the dimensional
5 regularization. All these things and the fact eq.(6) correctly gives the all known anomalies (both chiral and Weyl) were confirmed in [12]. Our task tD~ therefore reduces to the evaluation of tr < x e− x> 0 . { | | }|t It is well-known the anomaly can be fixed to some extent by the coho- mology of the corresponding symmetry operators. It is powerful in that no explicit calculation is required (hence we need not to worry about the regular- ization ambiguity) and only some consistency relations due to the symmetries are essential. Especially for the chiral anomaly, the anomaly is fixed except one overall coefficient. Let us do the cohomology analysis for the present conformal anomaly. Following Ref.[13] we consider the infinitesimal Weyl transformation in (3), δgµν =2α(x)gµν,δφ = 0 and regard the infinitesimal parameter α(x) as an anti-commuting one. 3 Then we can introduce the coboundary operator Σ which acts on the functional space of gµν and φ.
4 δ 2 Σ= d x2α(x)gµν (x) , Σ =0 , (7) Z δgµν(x) which is a nilpotent operator. It acts on the effective action Γ[g,φ] as
ΣΓ=¯h∆+ O(¯h2) , (8) where the right hand side is expanded w.r.t. the loop-number(¯h). In the anomaly, we usually consider only 1-loop part. ∆, after removing ”triv- ial terms”(see Appendix D), is defined as the anomaly in the cohomology. Because of the nilpotency property, Σ2 = 0, the term ∆ must satisfy the following consistency condition.
Σ∆=0 . (9)
We impose the following conditions on the ”basis” of the function space of ∆.
1. 4D general coordinate invariance.
2. Physical dimension is 4 ( Number of derivatives is 4). 3 Only from this paragraph to the end of this section, the Weyl transformation param- eter α(x) is infinitesimal and anti-commuting.
6 3. ”Ghost” number is 1. (This means we consider those terms which contain one α only.)
4. Symmetry of constant shift: P (φ) P (φ) + const . → The last item 4 is the speciality of the dilaton field. (Compare with the scalar field considered in [13].) For the familiar case P (φ) = φ, the trans- formation is the constant shift of the dilaton. The actions involving the dilaton (see (13), (29) and (54) in the following text) are essentially in- variant because the transformation changes the actions only by an over- all constant factor, which can be absorbed by the constant rescaling of the matter fields: f f const. This symmetry is very important be- → × cause it forbids, as the anomaly terms, the appearance of the terms like ,µ 2 2 ,µ 2 4 µνλσ 2 √g(P,µP ) P , √g(P,µP ) P , √gCµνλσC P , . ··· Let us see how the consistency condition (9) restricts ∆. In Table 1 we list all the possible terms, ∆i, which satisfy the above four conditions and Σ-transformed ones for each, Σ∆i.@ Before the consistency condition is imposed, the most general expression is given as
∆= ci∆i . (10) Xi From the consistency condition we obtain the following relations between coefficients.
i) c1 + c2 + c3 =0 . ii) 6c + c =0 , 2c 4c =0 , 2c +4c +2c =0 . 6 7 − 7 − 9 − 7 8 9 iii) 2c c =0 . 14 − 15 iv)c16 +3c19 =0 , 2c17 + c18 =0 . (11)
Other coefficients are arbitrary.
7 No ∆i Σ∆i µνλσ 2 1 dV RµνλσR α(x) 4 dV R α α µν ∇2 · 2 R dV Rµν R α(x) 4 R dV R α α 2 ∇ 2 · 3 R dV R α(x) 12R dV R α α 2 ∇ · 4 dVR ( R) α(x) R 0 ∇ ,µ ·2 5 R dV (P,µP ) α(x) 0 ,µ ,µ 2 6 R dV RP,µP α(x) 6 dVP,µP α α µν ,µ 2 ,µν ∇ · 2 ,µ 7 RdV R P,µP,να(x) dV (P,µP αR 2P P,µα,ν 2 P P α,µ)α 2 2 ∇ − 2 ,µ − ∇ · 8 R dV ( P ) α(x) R 4 dV P P α,µα ∇ ,µν· ,µν ∇ · 2 ,µ 9 R dVP,µνP α(x) dV ( 4P R P,µα,να +2 P P α,µα) 2 ,µ − ∇ · 10 dVR µ( P P ) α(x) R 0 ∇ ∇ ,µν· · 11 R dV µ(P P,ν) α(x) 0 ∇ 2 ,µ · 12 R dV (P,µP ) α(x) 0 ∇ ,µ ,ν· 13 RdV µ ν(P P ) α(x) 0 ∇ ∇ ,µ 2 · ,µ ,ν 14 R dV P,µP ( P ) α(x) 2 dVP,µP P,να α ,µ ,ν∇ · ,µ ,ν · 15 R dVP P P,µνα(x) R dVP,µP P,να α 4 − 2 ·,µ 16 R dV P α(x) R2 dVP,µ α α ∇ · − ∇ · 17 dV R 2P α(x) dV (6 2P 2α +2RP α,µ) α R ∇ · ∇ R ·∇ ,µ · 18 dV RµνP α(x) dV (3 2P 2α + RP α,µ) α R ,µν · R ∇ ·∇ ,µ · 19 dV (RP ,µ) α(x) 6 dV P ,µ 2α α R ∇µ · R − ∇ · ,µ R R Table 1 List of all possible terms ∆i which satisfy the four conditions in Sec.2 and their Σ-transformed ones(see eq.(7)). dV d4x√g. ≡ R R
Since we have 7 relations among 19 terms, 19 7=12 terms remain as the possible Weyl anomaly terms appearing in the− gravity-dilaton background system. Among the 12 terms, there are ”trivial” terms which can be freely adjusted by introducing local conter-terms in the action. In order to find all trivial terms, we list, in Table 2, all the possible local counterterms γi and the Σ-transformed ones for each, Σγi. The symmetries which define the counterterms are the same as those for ∆ except the condition 3. ”Ghost” number is 0.
8 No γi Σγi µνλσ 2 1 dV RµνλσR 4 dV α R µν ∇2 2 R dV Rµν R 4 R dV α R 2 ∇ 2 3 R dV R 12R dVα R 2 ∇ 4 dVR ( R) R 0 ∇ ,µ 2 5 RdV (P,µP ) 0 ,µ 2 ,µ 6 R dV RP,µP 6 dVα (P,µP ) µν 2 ,µ∇ ,µ ,ν 7 RdV R P,µP,ν dV α (RP,µP )+2 µ ν(P P ) 2 2 {∇ ∇2 ∇ ,µ } 8 R dV ( P ) R 4 dV α µ( P P ) ∇ ,µν − ∇,µν ∇ · 2 ,µ 9 R dVP,µν P 2 dVα R µ(2P P,ν P P ) 2 ,µ ∇ −∇ · 10 dVR µ( P P ) R 0 ∇ ∇ ,µν· 11 R dV µ(P P,ν) 0 ∇ 2 ,µ 12 R dV (P,µP ) 0 ∇ ,µ ,ν 13 dVR µ ν(P P ) 0 ∇ ∇ ,µ 2 ,µ ,ν 14 R dVP,µP ( P ) 2 dV α ν(P,µP P ) ,µ ,ν∇ − ∇ ,µ ,ν 15 R dVP P P,µν dVαR ν(P,µP P ) 4 ∇ 16 R dV P R 0 ∇ 17 dV R 2P 2 dVα 3 4P µ(RP ) R ∇ { ∇ −∇ ,µ } 18 dV Rµν P dVα 3 4P µ(RP ) R ,µν R { ∇ −∇ ,µ } 19 dV (RP ,µ) 0 R ∇µ R R Table 2 List of all possible local counterterms γi and their Σ-transformed ones. dV d4x√g. ≡ Therefore we conclude, from the cohomologyR analysis,R the Weyl anomaly in the gravity-dilaton background system is composed of four invariants (I0,I4,I2,I1), the Euler term √gE, seven trivial terms (J0,J2a,J2b,J2c,J2d,J3,J1). 4
δ ∆= b I + b √gE + c J + b I + b I + b I δα(x) 0 0 E 0 0 4 4 2 2 1 1 4 I0 ∆1 2∆2 + (1/3)∆3, √gE ∆1 4∆2 + ∆3, J0 ∆4, I4 ∆5, I2 ∼ − ∼ − ∼ ∼ ∼ (1/6)∆6 + ∆7 + (3/4)∆8 (1/2)∆9, I1 (1/2)∆17 + ∆18, J2a ∆12, J2b ∆11 ∆− , J ∆ , J ∆ −, J ∆ + 2∆∼ −, J 3∆ ∆ . ∼ ∼ − 10 2c ∼ 13 2d ∼ 11 3 ∼ 14 15 1 ∼ 16 − 19
9 +c2aJ2a + c2bJ2b + c2cJ2c + c2dJ2d + c3J3 + c1J1 . (12)
The definite expressions for the symbols (Ii,E,Ji) appear in the following text and in Appendix D. In the cohomology approach, there generally remains not a few unde- termined coefficients of conformal anomaly terms, compared with the chiral anomaly. In order to fix them completely, the ”full-fledged” dynamical cal- culation like the present approach is indispensable.
3 Trace anomaly for 4D dilaton coupled scalar
Let us start with the Lagrangian
1 4 P (φ) 2 S1[ϕ; g,φ]= d x√ge ϕ( + ξR)ϕ , (13) 2Z −∇ where ϕ,gµν and φ are the scalar, graviton and dilaton fields, respectively. The form of P (φ) is often chosen to be const φ, but we will keep a more × general form. We treat gµν and φ as the background fields and the scalar matter field, ϕ, as the quantum field. At the value of ξ = 1 , (13) is exactly − 6 (not ignoring total derivatives) conformally invariant. 5
2α(x) α(x) φ′ = φ , gµν ′ = e gµν , ϕ′ = e− ϕ . (15) 5 We cannot take the following actions (compare it with the 2D case[5]):
1 S [ϕ; g, φ]= d4x√geP (φ)( ϕ µϕ + ξRϕ2) , 2 2 ∇µ · ∇ Z 1 S ′[ϕ; g, φ]= S [ϕ; g, φ] d4x√g (eP (φ)ϕ µϕ) 2 2 − 2 ∇µ ∇ Z 1 = d4x√gϕeP (φ)( 2 P µ + ξR)ϕ . (14) 2 −∇ − ,µ∇ Z These two actions, in the P (φ) = const case, differ from (13) only by a total derivative. 1 For the general case, however, they are not conformally invariant even for ξ = 6 . The dilaton can select the coupling with the ordinary matter fields among some possibilities− which differ only by the total derivative term (surface term) at the P (φ) = const limit. ′ ~ P (φ) 2 µ ~ ~ † Note that the operator from S2 , D2 e ( P,µ + ξR), is hermitian (D2 = D2) 4 ~≡ † −∇ 4− ∇† ~ † for the natural definition: d x√g(D2ϕ) ϕ = d x√gϕ D2ϕ. R R
10 In order to normalize the kinetic term of the scalar field, we rescale it as 6
1 P (φ) Φ = e 2 ϕ . (16)
Adding a total derivative term, S1 changes to
1 4 2 1 ,µ 1 4 2 ,µ S = d x√gΦ P,µP + ξR Φ= S1 d x√g µ(Φ P ) .(17) 2Z {−∇ − 4 } − 4Z ∇ We do the second rescaling in order to make the integration measure BRST- invariant[14] with respect to (w.r.t.) 4D general coordinate symmetry.
Φ=˜ √4 gΦ . (18)
We can write S as 1 1 1 4 4 2 ,µ S = d xΦ˜D~ Φ˜ , D~ √g( ¯ ) , ¯ = ξR + P,µP . (19) 4 2Z ≡ −∇ − M √g M − 4 We generally have the freedom of the choice of surface terms (like (17)) and of rescaling the variables ( like (16) and (18) ). We cannot keep S1 as the 2 1 2 action, because the corresponding operator D~ ′ = √4 g( ¯ + P + −∇ − M 2 ∇ P ,µ ) 1 , is not hermitian due to the presence of the first derivative term. ∇µ √4 g We can define the functional space , in terms of the eigen functions of D~ , F as :
Du~ = λ u , = Φ=˜ c u c Complex, n =0, 1, 2, (20). n n n F { n n n| n ∈ ···} P The inner product between functions and the hermite conjugate of the oper- ator D~ are defined as
4 4 4 P (φ) < Φ˜ 1 Φ˜ 2 > d xΦ˜ †Φ˜ 2 = d x√gΦ† Φ2 = d x√ge ϕ†ϕ2 , | ≡ 1 1 1 R < D~ Φ˜ Φ˜ R> < Φ˜ D~ †Φ˜ >R . (21) 1| 2 ≡ 1| 2 6 A way to avoid the rescaling (16) is to take fully the dilaton coupling into the operator: P (φ) 2 † D~ 1 = e ( + ξR). This is, however, not hermitian(D~ 1 = (D~ 1) ) for the natural −∇4 † 4 † † 6 definition: d x√g(D~ 1ϕ) ϕ = d x√gϕ (D~ 1) ϕ. It should be compared with D~ 2 in the previous footnote. R R
11 Under this definition, the hermiticity of the operator D~ in (19) holds true. Taking the eigen functions u orthonormal, the integration measure is { n} given by
1 4 ˜ 1/4 2 P (φ)
1 1 2 1 1 2 1 ,µ 1 2 ,µ +(ξ + ) R + (ξ + )R RP,µP + (P,µP ) . (24) 6 −6∇ 2 6 − 4 24∇ } This result is consistent with [18] where dimensional regularization was used. The above result is not a conformal anomaly in the strict sense because D~ of (19) is not conformal invariant for the general value of ξ. At the classical 1 conformal limit ξ = 6 , the above result reduces to the ”true” conformal anomaly: −
√g 1 ,µ 2 Anomaly ξ= 1/6 = (P,µP ) | − (4π)2 {32 1 1 1 + (3C Cµνλσ E) 2R + 2(P P ,µ) , (25) 360 µνλσ − − 180∇ 24∇ ,µ }
12 where Cµνλσ and E are the conformal tensor(Appendix C) and the Euler term, respectively, which are defined by 1 C = R + (g R g R + g R g R ) λµνσ λµνσ 2 λν µσ − λσ µν µσ λν − µν λσ 1 + (g g g g )R, 6 λσ µν − λν µσ 1 C Cλµνσ = 2R Rµν + R Rµνλσ + R2 , λµνσ − µν µνλσ 3 1 E = R R ǫµνλσǫαβγδ = R2 + R Rµναβ 4R Rµν . (26) 4 µναβ λσγδ µναβ − µν In (25) we notice two conformal invariants: I √gC Cµνλσ,I √g(P P ,µ)2 , (27) 0 ≡ µνλσ 4 ≡ ,µ and two ”trivial” terms (Appendix D) J0 and J2a:
2 2 δ 4 J0 √g R , K0 √gR , gµν d xK0 =6J0 , ≡ ∇ ≡ δgµν (x)Z 2 ,µ ,µ δ 4 J2a √g (P,µP ) , K2a √gRP,µP , gµν d xK2a = +3J2a .(28) ≡ ∇ ≡ δgµν(x)Z The numbers of lower suffixes in I0,I4,J0,K0, etc, show the power numbers of P ( see Appendix D). We note that I4 and J2a are caused by the presence of the dilaton. Two terms J0 and J2a in (25) are arbitrarily adjusted by in- troducing finite counterterms, K0 and K2a respectively, in the original (local gravitational) action. This result says the conformal anomaly at ξ = 1 − 6 consists of two conformal invariants (I0,I4), the Euler term(√gE) and two trivial terms (J0,J2a). This confirms the general structure of the conformal anomaly discussed in [19, 20, 21, 10]. 2D case which is qualitatively the same is discussed in appendix E.
4 Trace anomaly for 4D dilaton coupled vec- tor
4.1 Basic Formalism In this section we consider the dilaton coupled photon system. The action is given by
13 4 1 P (φ) µν SV [A; g,φ]= d x√g( e FµνF ) , Fµν = µAν νAµ . (29) Z −4 ∇ −∇ Besides the 4D general coordinate symmetry, this theory has the local Weyl symmetry:
2α(x) gµν′ = e gµν , Aµ : fixed , φ : fixed , (30) and it has the Abelian local gauge symmetry:
A′ = A + Λ(x) , g : fixed , φ : fixed . (31) µ µ ∇µ µν We take the following gauge-fixing term:
4 1 P (φ) µ 2 Sgf [A; g,φ]= d x√g( e ( µA ) ) . (32) Z −2 ∇ The corresponding ghost lagrangian is
4 P (φ) µ Sgh[¯c,c; g,φ]= d x√ge c¯ µ c , (33) Z ∇ ∇ wherec ¯ and c are the anti-ghost and ghost fields respectively. They are hermitian scalar fields with Fermi statistics (Grassmannian variables). The total action S[A, c,c¯ ; g,φ] = SV + Sgf + Sgh is invariant under the BRST symmetry
δA = ξ∂ c , δc¯ = ξ µA , δc =0 , δφ =0 , δg =0 , (34) µ µ ∇ µ µν where ξ is the BRST parameter which is a global Grassmannian variable. Sgf breaks the Weyl symmetry (30), besides the gauge symmetry (31). Therefore we must take close care to define the conformal anomaly in gauge theory. It is highly non-trivial whether the general structure suggested by Ref.[20, 21] holds true in this case. We assign the following Weyl transformation toc ¯ and c [22]:
2α(x) c¯′ = e− c¯ , c′ = c , (35) which is chosen in the following way: i) the ghost action (33) is invariant under global Weyl transformation (α(x) is independent of x) ; ii) Sgf + Sgh
14 is invariant, up to a ”BRST-trivial” term, under infinitesimal local Weyl transformation. After rescaling the vector and ghost fields ( for normalizing the kinetic 1 P (φ) P (φ) term): e 2 Aµ = Bµ , e c¯ = ¯b, the actions can be expressed as
4 1 µ 2 λ ν SV + Sgf = d x[ √gB gµν +( )µν λ + µν B + total deri.] , 2 { ∇ N ∇ M } 4 ¯ 2 R Sgh = d x√gb c , ∇ 1 ( λ) = P δλ P δλ , R = R g (P P ,λ +2 2P ) , (36) N µν ,µ ν − ,ν µ Mµν µν − 4 µν ,λ ∇ Following Fujikawa[14], we take ( 4D general coordinate transformation sym- ˜ metry) BRST-invariant measure by taking B˜µ, ¯b andc ˜ instead of Bµ, ¯b and c respectively, as the path-integral variables 7 :
B˜ g1/8B (B˜µ g3/8Bµ) ,¯˜b √4 g¯b ,c˜ √4 gc , µ ≡ µ ≡ ≡ ≡ 4 1 ˜µ ~ ν ˜ SV + Sgf = d x[ B Dµ Bν + total deri.] , Z −2 ν 1/8 ν 2 λ ν ν 1/8 D~ = g δ +( ) + g− , µ − { µ ∇ N µ ∇λ Mµ } 1 4 ¯˜ ~ ~ 4 2 Sgh = d xb d c˜ , d √g , 4 −Z ≡ − ∇ √g exp Γ[g,φ] = B˜ ¯˜b c˜ exp(S˜[B,˜ ¯˜b, c˜; g,φ]) , { } Z D D D S˜[B,˜ ¯˜b, c˜; g,φ]= S[A, c,¯ c; g,φ] , (37) where Γ[g,φ] is the (Wilsonian) effective action. The Weyl transformation of (30) and that of the integration variables ˜ α ˜ ¯˜′ ¯˜ 2α (Bµ′ = e Bµ, b = b, c˜′ = e c˜) give us the Ward-Takahashi identity for the Weyl transformation: ˜ ˜ exp Γ[g′,φ]= B˜′ ¯b c˜′ exp S˜[B˜′, ¯b, c˜′; g′,φ] Z D D D ˜ ∂B′ ˜ ∂c˜′ ˜ = B˜ det ¯b c˜ det exp S˜[B˜′, ¯b, c˜′; g′,φ] ˜ Z D · ∂B ·D D · ∂c˜ ˜ ¯˜ α ν 4 2α 4 1 = B b c˜ det(e δµ δ (x y)) det(e δ (x y)) − Z D D D · − ·{ − } 7 The measure is also invariant under the BRST transformation w.r.t. the Abelian ˜ P/2 −1/8 1 −1 ¯˜ P/2 1/8 µ ˜ gauge symmetry: δBµ = ξe g (∂µc˜ 4 g ∂µg c˜) , δb = ξe g ( Bµ −′ · ∇ − 1 g−1g,µB˜ 1 P ,µB˜ ) , δc˜ =0 , det ∂(B˜′ , ¯˜b , c˜′)/∂(B˜ , ¯˜b, c˜) =1. 8 µ − 2 µ { µ ν } 15 ˜ ˜ ˜ ˜ δS δS δS exp S˜[B,˜ ¯b, c˜; g,φ]+ δB˜µ + δc˜ + δgµν . (38) × " δB˜µ δc˜ δgµν #
Considering the infinitesimal transformation (δgµν =2αgµν, δB˜µ = αB˜µ, δc˜ = 2αc˜) in the above equation, and regularizing the space delta-functions δ4(x − y) in terms of the heat-kernels, we obtain δΓ δS˜ δS˜ δS˜ Γ[g′,φ] Γ[g,φ]=2αgµν = α B˜µ +2α c˜ +2α gµν − δgµν * δB˜µ + * δc˜ + * δgµν + α tD~ ν 2α td~ +Tr ln e < x e− µ y > Tr ln e < x e− y > ,(39) { | | } − { | | } where t is the regularization parameter: t +0. The part of variation w.r.t. S˜ gives the ”naive” Ward-Takahashi identity,→ while the remaining two terms (terms with ”Tr”) give the deviation from it. (Compare this with the conformal gauge identities in gauge theories on curved space without the dilaton [17].) Therefore the last two terms can be regarded as the Weyl anomaly in this theory.
∂ ~ ν α tDµ T1 Tr ln e < x e− y> , ≡ ∂α { | | } α=0,t=0
∂ 2α td~ T2 Tr ln e < x e− y> . (40) ≡ − ∂α { | | } α=0,t=0
~ ν 1/8 ν 2 The anomaly formula for the differential operator Dµ = g δµ + λ ν ν 1/8 −λ { ∇ ( )µ λ + µ g− with arbitrary general covariants of and (tak- ingN the∇ heat-kernelM } regularization and the Fujikawa method)N is derMived in Appendix B and is given as
1 1 2 1 2 1 αβ Anomaly = 2 √g Tr[ D X + X + YαβY ] (4π) 6 2 12 1 µνλσ µν 2 2 +4 (RµνλσR Rµν R +0 R R) , × 180 − × −∇ 1 1 1 X ν = ν ( α) ν ( α ) ν δ νR, µ Mµ − 2 ∇αN µ − 4 N Nα µ − 6 µ ν 1 ν 1 ν ν (Yαβ)µ = ( α β)µ + ( α β)µ α β + Rαβµ , 2 ∇ N 4 N N − ↔ 1 1 (D X) ν = X ν + [ ,X] ν , (D2X) ν = 2X ν + [ α,D X] ν .(41) α µ ∇α µ 2 Nα µ µ ∇ µ 2 N α µ
16 Putting all explicit equations (36) into this formula, the vector contribution to the Weyl anomaly is finally given by
1 1 µνλσ µν 2 2 T1 = 2 √g ( 11RµνλσR + 86Rµν R 20R +6 R) (4π) 180 − − ∇ 1 1 5 1 + (P P ,µ)2 + 2(P P ,µ)+ ( 2P )2 P P ,µν {16 ,µ 12∇ ,µ 12 ∇ − 6 ,µν 2 1 + RµνP P RP P ,µ 3 ,µ ,ν − 6 ,µ } 1 4 1 1 2 ,µ ,ν 1 µν 1 µν + P (P,µν + gµν P ) P P + (R g R)P,µν . (42) {−3∇ − 6 2 ∇ · 3 − 2 } Terms in the first big bracket ( ) show the even power part of P (φ), while those in the second show the odd{ } power part. The ghost contribution to the Weyl anomaly is obtained from (23) with =0 as M
1 1 2 µνλσ µν 5 2 T2 = 2 √g 2 ( 6 R + RµνλσR Rµν R + R ) . (43) (4π) − × 180 − ∇ − 2 The factor 2 comes from the two fields of the ghosts c andc ¯ and their Fermi − statistics. There are no dilaton(P)-terms in the ghost part. T1 + T2 gives the final result of the Weyl anomaly. It is given by the Euler term(√gE), four Weyl invariants (I4,I2,I1,I0) and five trivial terms(J3,J2a,J2b,J1,J0) which are defined in Appendix D: 1 1 Anomaly = ( 31√gE + 18I + 18J ) (4π)2 {180 − 0 0 1 1 1 1 1 1 + I + I + I √g 4P J + J + J , (44) 16 4 2 3 1 − 3 ∇ − 12 3 12 2a 3 2b} except the term of ( 1/3)√g 4P . In the next subsection, we show this − ∇ term, combined with the contribution from the measure ambiguity, reduces to ( 2/15)J1. This result again confirms the general structure of the conformal anomaly− analysed in [19, 20, 21, 10]. A slightly generalized aspect of the above result is that the new terms, I2 and I1, are conformally invariant not exactly but only up to total derivative terms. See appendix D for details. The conformal anomaly for the ordinary photon-graviton theory (without the dilaton) is correctly reproduced for P (φ) = const[23, 24, 25].
17 4.2 Close treatment of Faddeev-Popov procedure and the integration variables In the previous section, the final result has the term √g 4P -term which is, ∇ by itself, not allowed by the cohomology analysis. We miss some contribution. First we notice one unsatisfactory treatment of the Faddeev-Popov procedure in the present ghost action (33). It originates from the ”insertion” of the following identity in the path-integral expression.
1 µ 4 2 P (φ) 1= ∆F δ(√ge µAΛ K) Λ , Z ∇ − D 1 1 1 4 P µ P P µ ∆F = det(√ge 2 µ ) = det( e− 2 ) det(√ge µ ) , (45) ∇ ∇ √4 g ∇ ∇ where K is a fixed constant and Λ(x) is the abelian local gauge parameter ˜ P µ (31).@∆F is the ”rigorous” F-P determinant whereas ∆F det(√ge µ ) is the one we have taken into account by (33). We must take≡ care of the∇ differ-∇ 1 1 P 1 P ence factor det( e− 2 ), especially its P -dependent factor ∆ det(e− 2 ) √4 g P ≡ for the present interest. ( We notice the similar situation occurred in the gauge parameter (in)dependence problem of the 2R anomaly-term in 4D vector model in the background metric (no dilaton)[34].∇ The careful treat- ment of such a factor as above solves the problem. We see, from the gauge- 1 P fixing term (32), e 2 corresponds to the ”gauge parameter”. ) Furthermore we notice such factors as ∆P , depending on the choice of the path-integral variables, can appear. For example, in eq.(36), such factors, depending on the choice of (Aµ, c¯) or (Bµ, ¯b), indeed appear. (This factor is important when the overall factor of the path-integral get involved with the problem. In the case of Ref.[34], the overall factor depends on the gauge parameter. It affects the Weyl anomaly. In the present case, it depends on the dilaton field P (φ).) 8 As far as the dilaton is treated as the bacground field, we have arbitrariness of such a factor. Therefore we may assume there exists a factor ∆ (α) det(eαP ) , with some number α, which multiplies the path-integral P ≡ we are considering. Let us evaluate its contribution to the Weyl anomaly. We consider the case P 1 which is the region where the perturba- | | ≪ tive definition can be safely applied. Then the determinant factor can be 8 In the string theory, the dilaton factor eP (φ) is regarded as a coupling. In the present problem the dilaton term behaves like the gauge parameter.
18 evaluated, in the heat-kernel regularization, as
αP (φ(x)) (4) td~ ln ∆P (α) = ln det e δ (x y) = α lim Tr P (φ(x)) < x e− y > − t +0 | | n o → n o 1 t(d~ P ) = α lim Tr < x e− − y> P 1 ,(46) t +0 → t | | | where d~ is given in (37). This trick of exponentiating the factor P was used in [26]. Let us evaluate the heat-kernel
P t(d~ P ) G (x, y; t) Tr < x e− − y > , ≡ | | ∂ ( + d~ P )GP (x, y; t)=0 . (47) ∂t − at the order of O(P 1) w.r.t. the dilaton field P , and of O(t) w.r.t. the proper time t. As for the background metric, it is sufficient to consider up to the first order of O(h) w.r.t. the weak-field h (g = δ + h , h 1). µν µν µν µν | µν | ≪ Their trace w.r.t. the space coordinates (x = y) is given by G1(x, x; t) and G2(x, x; t) appearing in eq.(24) and eq.(25) of [12] respectively. Taking 1 2 2 2 2 n =4, M = 4 ∂ h + P + O(h ), Wµν = hµν + O(h ), Nλ = ∂µhλµ + O(h ) − − P − in the reference we obtain the following things. (G (x, x; t) = G0(x, x; t)+ G (x, x; t)+ G (x, x; t)+ .) 1 2 ··· (i) G1(x, x; t) From (24) of [12], we obtain
1 2 1 4 1 w − 4(1−r) 4 d w dr 2 e (4π) t Z Z0 (1 r)r 2 { − } 1 4 α β γ δ w 1 t 4 ∂ ∂ ∂ ∂ P (√t) w w w w e− 4r = ∂ P. (48) ×4! α β γ δ · (4π)2 60
(ii) G2(x, x; t) From (25) of [12], the relevent part of ∂3h ∂P is given by · 1 k 2 1 4 4 1 v d v d u dk dl e− 4(1−k) (4π)6 0 0 (1 k)(k l)l 2 Z Z Z Z { − − } (v−u)2 1 (√t)3 ∂ ∂ ω 4(k−l) α β γ ∂ωP √tv e− ∂α∂β∂γ ( hµν ) u u u × " ( t 3! − · ∂uµ ∂uν
2 2 1 (√t) α β ∂ 1 2 α u + ∂α∂β( ∂λhµλ) u u + √t∂α( ∂ h) u e− 4l √t 2! − · ∂uµ −4 · )
19 3 2 1 (√t) α β γ ∂ ∂ 1 (√t) α β ∂ + ∂α∂β∂γ ( hµν ) v v v + ∂α∂β( ∂λhµλ) v v ( t 3! − · ∂vµ ∂vν √t 2! − · ∂vµ 2 1 (v−u) u2 2 α 4(k−l) ω +√t∂α( ∂ h) v e− ∂ωP √tu e− 4l −4 · × · # t 1 2 1 = 2 ∂µ∂ hνν ∂µP + ∂µ∂ν ∂λhνλ ∂µP + other terns .(49) (4π) −40 · 30 · Making use of the weak-field expansion expressions 1 4P = ∂4P + ∂ ∂2h ∂ P ∂2∂ h ∂ P + O(h2) , ∇ 2 µ · µ − µ µν · ν R P ,λ = ∂ ∂2h ∂ P ∂ ∂ ∂ h ∂ P + O(h2) , (50) ∇λ · µ · µ − µ ν λ νλ · µ we demand the requirement of the cohomolgy analysis, that is, equating the following quantity
1 1 4 1 1 4 1 P ( ) P + ln ∆P (α)= ( ) P + α lim Tr G (x, y; t) P 1 2 2 t +0 (4π) −3 ∇ (4π) −3 ∇ → t | 1 α 1 α 1 α = (( )∂4P ( + )∂ ∂2h ∂ P + ∂ ∂ ∂ h ∂ P (4π)2 60 − 3 − 40 6 µ · µ 30 µ ν λ νλ · µ +other terms) ,(51) with the trivial term
β J = β√g(3 4P λ(RP )) × 1 ∇ −∇ ,λ 4 1 2 = β 3∂ P + ∂µ∂ h P,µ + ∂µ∂ν∂λhνλ P,µ + other terms , (52) 2 · · we find 1 2 α = 4 , β = ( ) . (53) − (4π)2 × −15
The consistency with the cohomology determines the dilaton-depending over- all factor of the path-integral.
20 5 Trace anomaly for 4D dilaton coupled spinor
The dilaton can be introduced into the (Dirac) spinor-gravity system keeping the Weyl invariance as follows:
¯ 1 4 P (φ) ¯ ¯ ¯ µ ¯ µ S[ψ, ψ; g,φ]= d x√ge ψi↔/ ψ , ψ↔/ ψ = ψ(γ µψ) (ψ←−µγ )ψ , 2Z { ∇ } ∇ ∇ − ∇ µ ¯ ¯ / = γ µ , µψ =(∂µ + ωµ)ψ , ψ←−µ = ψ(←−∂ µ ωµ), ∇ ∇ ∇ ∇ − 1 ω a = (Γλ e a ∂ e a)eν , ω = σabω .(54) µ b µν λ − µ ν b µ 2 µab It is invariant under the Weyl transformation:
3 3 ¯ 2 α ¯ 2 α a α a ψ′ = e− ψ , ψ′ = e− ψ , eµ ′ = e eµ , φ′ = φ . (55)
We can absorb all P (φ) dependence by the rescaling of fields:
1 P (φ) 1 P (φ) e 2 ψ = χ , e 2 ψ¯ =χ ¯ , 1 1 1 S = d4x√gχi¯ / χ = d4xχ˜¯√4 gi / χ˜ , (56) ↔ ↔ 4 2Z ∇ 2Z ∇ √g where χ˜¯ = √4 gχ¯ andχ ˜ = √4 gχ. Therefore the dilaton does not change the conformal anomaly in the spinor-gravity theory as far as we take the appropriate measure: χ˜¯ χ˜ = (√4 geP/2ψ) (√4 geP/2ψ¯). D D D D 6 Induced Action for Dilatonic Wess-Zumino model
In the present section we apply the previous results to the dilatonic Wess- Zumino (WZ) model and discuss the conformal anomaly and the anomaly induced action. The model may be constructed following the book by Wess and Bagger[27]. We suppose that the model is coupled with the external gravitational multiplet and the external dilaton (chiral) multiplet Ψ. Then using the notation of above book we may write the conformally invariant action:
2 1 P (Ψ) P (Ψ†) S = d Θ 2 [ ( ¯ ¯ +8R) Φ†(e Φ) + e Φ†Φ ] , (57) Z E −16 DD { } 21 where Φ = A + √2Θχ + ... and Ψ = φ + . Choosing only the bosonic background for Ψ and eliminating the auxiliary··· fields leads to the following component form of WZ action.
P (φ) 2 1 i µ µ L = e e A A∗ RAA∗ (χσ χ¯ +χσ ¯ χ) , (58) {− ∇ − 6 − 2 Dµ Dµ } where we take the choice φ∗ = φ for the dilaton. A is the complex scalar and χ is the Majorana spinor. The conformal anomaly for the system may be found as follows
1 T = 2 bI0 + b′√gE + b′′J0 + aI4 + a′J2a , (4π) { } 1 2 1 1 2 1 1 1 1 a = 32× = 16 , a′ = 24× = 12 , b = 120 (2+6 2 )= 24 , 1 1 1 1 × 1 1 b′ = (2+11 )= , b′′ = (2+6 )= . (59) − 360 × 2 − 48 − 180 × 2 − 36 b′′ and a′ are ambiguous as they may be changed by some finite counterterms in the gravitational action. The nonlocal effective action may be found by integrating the Weyl- transformed anomaly equation: 9 1 δ T (e2σg )= W (e2σg ). (60) µν 2 δσ µν The final expression for the anomaly induced effective action W is given as
1 4 4 b′ 2 1 2 W (gµν)= 2 d xd y bI0 + aI4 + (√gE + J0) xG(x, y) (√gE + J0) y (4π) { 2 3 } { 4 3 } 1 4 1 2 a′ R + d x (b′′ b′)K + K , (61) (4π)2 { 12 − 3 0 6 2a} R where W is in a covariant but non-local form via 2 1 (√g∆) G(x, y)= δ4(x y) , ∆=( 2)2 2Rµν + R 2 ( µR) .(62) x − ∇ − ∇µ∇ν 3 ∇ − 3 ∇ ∇µ Here ∆ is the conformally invariant 4-th order differential operator (Appendix D). For the case a = a′ = 0, the result (61) coincides with Ref.[28]. Note that it is the effective action (61) that may serve as a starting point for constructing IR sector of the dilatonic gravity in the analogy with the purely gravity case of [31, 32]. 9 See [28, 29, 30] for the purely gravitational case and [18] for the dilaton-gravitational case.
22 From another point using eq.(61) one can study the backreaction problem for the dilatonic WZ model. Here new possibilities are expected to appear in constructing the inflationary universe induced by the model. We hope to return to this question in more detail in future.
7 Conclusion
We have shown how the dilaton affects the conformal anomaly in various gravity-matter theories. The dilaton has the same status as the gravitational field. We summarize the results of this paper by writing the anomaly TS, TV and TF for the scalar, the vector and the Dirac fermion respectively:
1 1 1 TS = 2 ( √gE +3I0)+ I4 + Trivial Terms , (4π) { 360 − 32 } 1 1 1 1 TV = 2 ( 31√gE + 18I0)+ I4 + I2 + I1 + Trivial Terms , (4π) { 180 − 16 3 } 1 1 11 TF = 2 ( √gE +9I0) + Trivial Terms . (63) (4π) { 180 − 2 }
For the system with NS scalars, NF Dirac fermions and NV vectors, the conformal anomaly is given by
1 Anomaly = 2 bE√gE + b0I0 + b4I4 + b2I2 + b1I1 + Trivial Terms , (4π) { } b = 1 ( N 62N 11N ) < 0 , b = 1 (N + 12N +6N ) > 0 , E 360 − S − V − F 0 120 S V F b = 1 (N +2N +0 N ) > 0 , b =0 N + N +0 N > 0 , 4 32 S V × F 2 × S V × F b =0 N + 1 N +0 N > 0 . (64) 1 × S 3 V × F This is the dilaton generalization of a result in Ref.[20]. ”Trivial terms” depend on the choice of regularization, measure and finite counterterms[5]. Their meaning is not very clear ( at least, in the perturbative approach). 10 On the other hand, the coefficients b’s are clearly obtained and depend only on the content of the matter fields. They remind us of the critical indices in condensed matter physics. In Sec.5 we have presented the anomaly induced action for the dilatonic Wess-Zumino model by integrating the conformal anomaly equation w.r.t. the Weyl mode. It is known that some anomaly equations like those of QED 10 The gauge (in)dependence of the ”trivial terms” was analysed in Ref.[33] and the gauge independence was explicitly shown in Ref.[34].
23 or λφ4 theory on the curved space (with account of matter interaction) can not be integrated because of the presence of R2-term in the anomaly [28]. Recently, in Ref.[9] it is noted that the anomaly for the free photon on the dilatonic curved space (44), cannot be integrated because of the presence of the new type (generalized) conformal invariants I2,I1a and I1b. In other words, a covariant anomaly induced effective action cannot be constructed for a dilaton coupled vector. It indicates some speciality of the gauge theory in comparison with the scalar and fermion fields. It could be possible to construct only non-covariant anomaly-induced effective action in the dilaton- coupled vector case. Probably, such an action still could lead to a correct conformal anomaly like in the case with Chern-Simons invariants. It is also interesting to note that our results maybe well applied to SUSY dilaton coupled theories like Wess-Zumino model (for two-dimensional case, see [38]). If string theory or other alternative theory (like M-theory) takes over local field theory, the latter formalism should show some theoretical break- down at some scale such as the Planck mass. At present some pathological phenomena, such as the entropy loss problem in the black hole geometry could look like such a breakdown. We note here that the conformal anomaly is considered to be a key factor in the Hawking radiation phenomenon. In the transitive period from local field theory to string field theory, ”interme- diate” fields such as the dilaton are thought to play an important role in understanding new concepts involved in the Planck mass physics. The Weyl anomaly occurs quite generally for most ordinary conformally invariant field theories. It shows that the quantization of fields necessarily breaks the scale invariance. It is always related with the ultraviolet regularization. On the other hand it is frequently suggested that string theory could predict some minimal unit of length in space(-time). The effect of such a unit length on its field theory limit could be related to the conformal anomaly phenomenon. We hope the present results are helpful in studying of such problems.
24 Appendix A. Anomaly Formula for Scalar Theories on Curved Space
In this appendix we derive an anomaly formula for the general differential operator 1 D~ = √4 g( 2 1 ) , (65) −∇ · −M √4 g which frequently appears in the scalar theories on curved space. 1 is the unit matrix with the size of the field indices: (1) = δ ; i, j = 1, 2, , N . We ij ij ··· s proceed in three steps. 11 i) Flat space For the case of the flat space : g = δ the operator reduces to ∂2 µν µν − −M and the anomaly is known [12, 36] as 1 1 1 Anomaly = Tr ( ∂2 + 2) . (66) (4π)2 6 M 2M where ”Tr ” means ”trace” over the field suffixes i: Tr = Ns . M i=1 Mii ii) General curved space P From the general covariance the most general form which reduces to (66) in the flat space can be expressed as √g 1 1 Anomaly = Tr 2X + X2 (4π)2 {6∇ 2 +(a R Rµνλσ + a R Rµν + a R2 + a 2R) 1 , 1 µνλσ 2 µν 3 4∇ · } X = + c R, (67) M 1 where a a and c are some constants to be determined. 1 ··· 4 1 iii) Determination of constants 1 First we consider the case = 6 R which corresponds to the case:f(φ) = 1,ξ = 1 in Sec.2. The aboveM result (67) says − 6 1 1 N =1 , = R , X =( + c )R, s M 6 6 1 11 This approach was taken in the derivation of the counterterm formula[35].
25 √g 1 1 2 Anomaly = 2 ( + c1)+ a4 R (4π) {6 6 }∇ µνλσ µν 1 1 2 2 +a1RµνλσR + a2Rµν R + ( + c1) + a3 R . (68) {2 6 } Comparing this with the well-established result (say, eq.(40) of [12])
√g 1 Anomaly = ( 2R + R Rµνλσ R Rµν ) . (69) (4π)2 180 −∇ µνλσ − µν we obtain the first set of constraints on the constants. 1 1 a = , a = − , 1 180 2 180 1 1 1 1 1 ( + c )+ a = − , ( + c )2 + a =0 . (70) 6 6 1 4 180 2 6 1 3 Next we consider the case of the weak gravity (g = δ +h , h 1) µν µν µν | µν|≪ for the general operator (65) in order to fix the value of c1. The operator is expanded as
D~ = 1∂2 + 1(h ∂ ∂ + ∂ h ∂ + 1 ∂2h)+ O(h2) − −M µν µ ν µ µλ λ 4 1∂2 W ∂ ∂ N ∂ M. ≡ − − µν µ ν − λ λ − W = h + O(h2) , N = ∂ h + O(h2) , M = 1 ∂2h + O(h2)(71). µν − µν λ − µ µλ M − 4 2 2 2 Now we focus on the term (c1/(4π) )√g R =(c1/(4π) ) (∂ h ∂µ∂ν hµν )+ M M − O(h2) in (67). The conformal anomaly from (71) is independently obtained by the result of [12] ((38),Table I) taking Wµν , Nλ, M in the reference as given above: 1 1 1 1 1 Tr ( M∂2W + M∂ ∂ W M∂ N + M 2) (4π)2 −12 µµ 3 µ ν µν − 2 µ µ 2 1 1 1 1 1 − Tr ( ∂2h ∂ ∂ h + O(h2)) = √g(− )Tr R. (72) ∼ (4π)2 M 6 − 6 µ ν µν (4π)2 6 M
This shows 1 c = . (73) 1 −6
26 (70) and (73) fix all constants. Finally we obtain the anomaly formula for (65) as
√g 1 1 Anomaly = Tr 2X + X2 (4π)2 {6∇ 2 1 + (R Rµνλσ R Rµν +0 R2 2R) 1 , 180 µνλσ − µν · −∇ · } 1 X = R 1 . (74) M − 6 · As noted in the text, this result ( and (41) to be derived in the next appendix ) is the same as the counterterm formula (say of ref. [15, 16, 17]) where dimensional regularization was taken.
Appendix B. Anomaly Formula for Vector Theories on Curved Space
In this appendix we derive the anomaly formula for the operator
ν 1/8 ν 2 λ ν ν 1/8 D~ = g δ +( ) + g− , (75) µ − { µ ∇ N µ ∇λ Mµ } which generally appears for the vector fields on the curved background. λ and are general covariants. We proceed in the similar way as in App.A.N M i)Flat Space For the case of flat space gµν = δµν , the formula appears in [12, 36]:
D~ ν = δ ν∂2 ( λ) ν∂ ν , µ − µ − N µ λ −Mµ 1 1 X ν = ν (∂ λ) ν ( λ) ν , µ Mµ − 2 λN µ − 4 NλN µ 1 1 (Y µν ) β = (∂µ ν ∂ν µ) β + [ µ, ν] β , α 2 N − N α 4 N N α 1 DµX β = ∂µX β + [ µ,X] β , α α 2 N α 1 1 1 1 Anomaly = Tr [ D2X + X2 + Y Y µν ] , (76) flat (4π)2 6 2 12 µν
27 where the upper and lower indices do not have meaning in present case µν ν (gµν = g = δµ are all the Kronecker’s delta), but only do have a notational continuity with the next general case. ii) Curved Space The anomaly formula can be first expressed as the most general form which reduces to (76) at the flat space limit: 1 1 X ν = ν ( λ) ν ( λ) ν + c δνR, µ Mµ − 2 ∇λN µ − 4 NλN µ 1 µ 1 1 (Y µν) β = ( µ ν ν µ) β + [ µ, ν] β + c Rµν β , α 2 ∇ N −∇ N α 4 N N α 2 α 1 DµX β = µX β + [ µ,X] β , α ∇ α 2 N α √g 1 1 1 Anomaly = Tr D2X + X2 + Y Y µν (4π)2 {6 2 12 µν +(a R Rµνλσ + a R Rµν + a R2 + a 2R) 1 , (77) 1 µνλσ 2 µν 3 4∇ · } where c1,c2, a1 4 are some constants to be determined. − iii) Determination of constants iiia) First we consider a special case of ( λ) ν =0, ν = R ν , which is N µ Mµ µ a special case of f(φ) = 1 in (29) and is the ordinary photon-gravity theory. Comparing the result from (77) ν ν ν µν β µν β Xµ = Rµ + c1δµR, (Y )α = c2R α , √g 1 1 Anomaly = 2(R +4c R)+ (R ν + c δνR)2 (4π)2 {6∇ 1 2 µ 1 µ (c )2 + 2 Rµν βR α +4a R Rµνλσ +4a R Rµν +4a R2 +4a 2R ,(78) 12 α µνβ 1 µνλσ 2 µν 3 4∇ } with the well-established result (say, eq.(2.14) of [33]) √g 11 43 1 1 Anomaly = R Rµνλσ + R Rµν R2 + 2R ,,(79) (4π)2 {−180 µνλσ 90 µν − 9 30∇ } we obtain 1 11 1 43 (c )2 +4a = , +4a = , − 12 2 1 −180 2 2 90 1 1 1+4c 1 (2c +4c 2)+4a = , 1 +4a = . (80) 2 1 1 3 −9 6 4 30
28 We give here a list of the weak-field expansion of the operator (75) for the use in iiib) and iiic).
g = δ + h , h 1 , µν µν µν | µν|≪ D~ β = δβ∂2 (W ) β∂ ∂ (N ) β∂ M β , (W ) β = δ βh + O(h2) , α − α − µν α µ ν − µ α µ − α µν α − α µν 1 (N ) β =( µ) β δβ(∂ h ∂ h) (∂ h + ∂ h ∂ h )+ O(h2) , µ α N α − α λ λµ − 4 µ − µ αβ α µβ − β µα 1 1 M β = β δβ∂2h ∂ (∂ h + ∂ h ∂ h ) α Mα − 8 α − 2 µ µ αβ α µβ − β µα 1 1 ( λ) β∂ h ( λ) τ (∂ h + ∂ h ∂ h )+ O(h2) .(81) −8 N α λ − 2 N α λ βτ τ λβ − β λτ
λ ν ν ~ ν iiib) Second we consider the case ( )µ = 0 with a general µ : Dµ = 1/8 ν 2 ν 1/8 N M g δµ + µ g− . We focus on the RTr term. The formula (77) gives− { ∇ M } M c 1 RTr . (82) (4π)2 M
On the other hand, the anomaly can be independently obtained by the weak- field expansion (81) and the anomaly formula eq.(38) and TABLE I of ref.[12]: 1 1 1 1 1 Tr ( M ∂2W + M ∂ ∂ W M ∂ N + M 2) (4π)2 −12 · µµ 3 · µ ν µν − 2 · µ µ 2 1 1 1 1 ( ∂2h + ∂ ∂ h ) Tr = − (R + O(h2))Tr ,(83) ∼ (4π)2 × 6 − µ ν µν × M (4π)2 × 6 M and we obtain 1 c = . (84) 1 −6
iiic) With the purpose of determining c2, we consider the = 0 case λ ~ β 1/8 β 2 λ β 1/8 M with a general : Dα = g δα +( )α λ g− . We focus on the ( ) RµνβαNterm in the− anomaly.{ ∇ FromN (77)∇ we} have NµNν αβ c + 2 ( ) Rµνβα . (85) 12 NµNν αβ
29 We can derive the above result independently in the weak gravity expansion (81). Eq.(43) of [36] gives, as the corresponding ∂∂h part, NN 1 1 Tr Y Y 1 ( 1 ) 1 (∂ ∂ h ∂ ∂ h µ ν)( µ ν) α (4π)2 12 µν µν ∼ (4π)2 × − 12 × 2 µ α νβ − µ β να − ↔ N N β 1 (+ 1 )R β( µ ν) α , (86) ∼ (4π)2 × 12 µνα N N β 1 where Yµν ∂µAν ∂νAµ + [Aµ, Aν], Aµ 2 (δµν + Zµν)Nν and (δµν + W )(δ +≡Z )= δ− 1. Therefore we obtain≡ µν νλ νλ µλ ·
c2 = +1 . (87)
Finally we obtain 1 1 1 1 a =+ , a = , a =0 , a = , c = , c = +1 , (88) 1 180 2 −180 3 4 −180 1 −6 2 and the general formula is given by (41).
Appendix C. Weyl Transformation
We list here some formulae of Weyl transformation for various fields. Only in this appendix the space(-time) is n-dimensional. i) Definition
1 Rλ = ∂ Γλ +Γλ Γτ ν σ , Γλ = gλσ(∂ g + ∂ g ∂ g ) , µνσ ν µσ τν µσ − ↔ µν 2 µ σν ν σµ − σ µν R = Rλ , R = gµνR , g = +detg , A = ∂ A Γλ A .(89) µν µνλ µν µν ∇µ ν µ ν − µν λ
ii) Weyl transformation of gravitational fields
2α(x) nα λ λ λ λ ,λ g′ = e g , g = e √g , Γ ′ Γ = δ α + δ α g α , (90) µν µν ′ µν − µν ν ,µ µ ,ν − µν q
30 ,λ λσ where α,µ = µα(= ∂µα),α = g α,σ. This is the finite transformation, not the infinitesimal∇ one:
Rλ ′ Rλ = δλ(α α α + g α α,τ )+ g (α,λ α,λα ) (ν σ) , µνσ − µνσ σ ,µν − ,µ ,ν µν ,τ µν ,σ − ,σ − ↔ 2 ,λ Rµν ′ Rµν = gµν α +(n 2)(α,µν α,µα,ν + gµνα,λα ) , − 2α ∇ − 2 − ,µ R′ = e− R + 2(n 1) α +(n 1)(n 2)α α ,(91) { − ∇ − − ,µ } where α = α. ,µν ∇ν∇µ iii) Weyl tensor The Weyl tensors defined below are invariant under the Weyl transformation: 1 Cλ = Rλ + (δλR + g Rλ ν σ) µνσ µνσ n 2 ν µσ µσ ν − ↔ −1 + (δλg ν σ)R, (n 2)(n 1) σ µν − ↔ − − λ ′ λ τ 2α C µνσ = C µνσ , Cλµνσ = gλτ C µνσ , Cλµνσ′ = e Cλµνσ . (92)
iv) Scalar
n−2 α(x) n−2 α(x) n 2 ϕ′ = e− 2 ϕ , ( ϕ)′ = e− 2 ( ϕ − α ϕ) , ∇µ ∇µ − 2 ,µ 2 2 n+2 α(x) 2 n 2 2 (n 2) ,µ ( ϕ)′ = e− 2 ϕ − ϕ α − α α ϕ . (93) ∇ {∇ − 2 ∇ − 4 ,µ } The well-known conformal coupling is recognized from the relation:
n 2 ′ n 2 √g(ϕ 2ϕ + − Rϕ2) = √g(ϕ 2ϕ + − Rϕ2) . (94) " ∇ 4(n 1) # ∇ 4(n 1) − − v) Dilaton
2 2α 2 ,λ φ′ = φ , ( φ)′ = φ(= ∂ φ) , ( φ)′ = e− φ +(n 2)α φ .(95) ∇µ ∇µ µ ∇ {∇ − ,λ} We note the dilaton and the scalar transform in the same way only in the 2D space:
µ 2α µ ( µφ φ)′ = e− µφ φ , ∇ ·∇ ∇ ·∇,λ ( φ)′ = φ (α φ + α φ g α φ ) . (96) ∇µ∇ν ∇µ∇ν − ,µ ,ν ,ν ,µ − µν ,λ 31 µ n/2 Here we see √g( µφ φ) is conformal invariant in n-dim space. In the text the general∇ form·∇ of the dilaton coupling eP (φ) is used instead of const φ e × . We may substitute P (φ) for φ in the above formulae (e.g. P,µν′ = P (α P + α P g α,λP )). ,µν − ,µ ,ν ,ν ,µ − µν ,λ
Appendix D. Weyl Invariants and Trivial Terms
We consider 4D space in this appendix. In the present model we have the following conformal invariants made only of the metric and the dilaton field:
µνλσ ,µ 2 I0 √gCµνλσC , I4 √g(P,µP ) , ≡µν 1 ,µ 3≡ 2 2 1 ,µν I2 √g(R P,µP,ν 6 RP,µP + 4 ( P ) 2 P P,µν ) , ≡ µν 1 µν− ∇ µν −1 µν I1 √g(R g R)P,µν = √g ν (R g R)P,µ , (97) ≡ − 2 ∇ { − 2 } where the numbers in the lower suffixes show the numbers of powers P . They transform under the finite Weyl transformation as
I0′ = I0 , I4′ = I4 , µ ,ν I2′ I2 =2√g (α P,µP,ν) , ν −,µ 2 ∇ ,µ ,µ I1′ I1 = √g 2(P,µνα P να) (2α α,νP,µ + α,µα P,ν) (98). − ∇ { −∇ ·∇ − }
Two terms I2 and I1, which are conformally invariant up to total derivative terms, may be called generalized conformal invariants. They lead to strange consequences, in comparison with standard conformal invariants, in construc- tion of the anomaly induced effective action. As the quantities with the same properties, we note the Euler term, √gE, defined in (26) and a trivial term, J0, defined below.
µ ν ,ν 2 ,ν (√gE)′ √gE =4√g Rα,µ 2Rµ α,ν µ(α,να )+2α,µ α +2α,να α,µ , 2 − ∇ µ{ −2 −∇,ν 1 2 ∇ ,ν } 3 (J0′ J0)=4√g µ( α + α,να ) α,µ( 3 R +2 α +2α,να ) , − ∇ {∇2 ∇ 2− ∇ } (√gE + J0)′ (√gE + J0)=4√g∆α , (99) 3 − 3 where √g∆ is the conformally invariant 4-th order differential operator de- fined in eq. (62) of the text.
32 Among the Weyl anomaly terms there are those terms which can be obtained by the infinitesimal Weyl transformation of local (counter)terms composed only of the background fields (gµν,φ). They are called ”trivial” (anomaly) terms [21]. ( In the mathematical (cohomology) terminology, it is called coboundary [37]). To obtain them, it is sufficient to consider the infinitesimal transformation(α(x) 1) in the appendix C: ≪ 2 2 δ 4 J0 √g R, K0 √gR , gµν d xK0 =6J0 , ≡ ∇ ≡ δgµν (x) 2 ,µ ,µ δ 4 J2a √g (P,µP ) , K2a √gRP,µP , gµν R d xK2a = +3J2a , ≡ ∇ ≡ δgµν (x) µν ,µν 2 2 ,µν ,µ 2 J2b √g( R P,µP,ν + P,µνP ( P ) )= √g µ(RP P,ν P P ) , ≡ − ,µν 1 2 − 2∇ δ ∇4 − ∇ K2b √g(P P,µν + ( P ) ) , gµν d xK2b = +2J2b , ≡ 2 ∇ δgµν (x) ,µ ,ν µν 1 ,µ J2c √g µ ν(P P ) , K2c √g(R P,µR P,ν RP,µP ), ≡ ∇ ∇ ≡ − 6 g δ d4xK =+J , µν δgµν (x) 2c 2c ,µν ,µν 1 2 2 J2d √g µ(P P,ν) , KR 2d √g(P,µνP ( P ) ) , ≡ ∇ ≡ − 2 ∇ g δ d4xK = +2J , µν δgµν (x) 2d 2d 2 ,µ ,µ ,ν ν ,µ J3 √g( P P,µP +2R P P P,µν)= √g (P,νP,µP ) , ≡ ∇ · ,µν δ 4 ∇ 1 K3 √gP P,µP,ν , gµν d xK3 = J3 , ≡ δgµν (x) 2 4 λ 2 δ 4 J1 √g(3 P (RP,λ)) , K1 √gR RP,gµν d xK1 = J(100)1 . ≡ ∇ −∇ ≡ ∇ δgµν (x) R J’s are trivial terms, while K’s are the corresponding counterterms. We note all trivial terms are total derivative ones. But total derivative terms are not 12 always trivial ones. I1 is such an example .
Appendix E. Conformal Anomaly for 2D Dilaton Coupled Scalar
For completeness we write below the conformal anomaly for 2D dilaton
12 I1 looks like the Euler term in the sense that its Weyl transformation is similar, as pointed out below (98), and it is the surface term which depends only on the topology.
33 coupled scalar with the action:
1 2 2φ 2 S = d x√ge− ( ϕ) . (101) 2 Z ∇ The notation is the same as in Sec.2 of the text. The calculation of the corresponding trace anomaly has been done actually in Ref.[3] (see later works[2, 4, 5, 6]) with the following result
√g T = R 6( φ)2 +6 2φ . (102) 24π {− − ∇ ∇ } The structure of the above anomaly consists of the Euler term, √gR, a trivial term, √g 2φ, and a conformally invariant term, √g( φ)2. So the structure is the same∇ as 4D dilaton coupled scalar in Sec.2 of the∇ text. The analysis given in above works leads to different values for the coeffi- cient of the trivial term. In particulary, the result of ref.[2] does not coincide with the calculations of refs.[3, 4, 5, 6]. The discrepancy occurs because of the difference in the choice of the ultraviolet regularization, the system oper- ator and the measure[5]. The most crucial point is, as far as we are studying the anomaly problem in the perturbative quantization of the matter fields only, we have no control over the purely gravity-dilaton sector. It means the corresponding coefficient appears to be ambigious. We have the freedom to introduce a finite counterterm √gRφ in the local gravity-dilaton action. (As for the the most general form of such a local action, see Ref.[39].) This changes the coefficient of the trivial term freely due to relation:
δ 2 2 gµν d x√gRφ = √g φ . (103) δgµν(x) Z ∇ Hence, the coefficient of the trivial term is ambigious (actually, it is defined by the normalization condition). Looking at the situation in the 2D black hole physics, the radiation amplitude is determined by the conformal anomaly equation and conservation of the energy- momentum tensor. The trivial term equally contributes to its determination. If we regard the radiation amplitude to be physical, it appears that we have to treat the the matter-gravity-dilaton system both nonperturbatively and on an equal footing with all fields.
34 Acknowledgements
The authors thank K.Fujikawa and R.Kantowski for reading the manuscript and some remarks. SI thanks R.Endo for some comments. SDO is grateful to I.Buchbinder and S.Nojiri for related discussions.
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37