6D trace anomalies from quantum mechanical path integrals

6D trace anomalies from quantum mechanical path integrals

Fiorenzo Bastianelli*

Dipartimento di Fisica, Universita` di Bologna, Italy and INFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy

Olindo Corradini†

C. N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794-3840

共Received 10 November 2000; published 9 February 2001兲

We use the recently developed dimensional regularization 共DR兲 scheme for quantum mechanical path integrals in curved space and with a finite time interval to compute the trace anomalies for a scalar field in six dimensions. This application provides a further test of the DR method applied to quantum mechanics. It shows the efficiency in higher loop computations of having to deal with covariant counterterms only, as required by the DR scheme.

DOI: 10.1103/PhysRevD.63.065005 PACS number共s兲: 11.10.Kk, 03.65.Db, 04.62.⫹v

I. INTRODUCTION

Quantum mechanical共QM兲 path integrals have been use-fully applied to the computation of chiral 关1,2兴 and trace 关3,4兴 anomalies. In these applications the anomalies are iden-tified as certain quantum field theory 共QFT兲 path integral Jacobians 关5兴, first reinterpreted as quantum mechanical traces and then given a path integral representation.

The topological character of chiral anomalies explains the relative easiness of computing the corresponding QM path integrals: the interpretation of chiral anomalies as indices of certain differential operators shows how the leading semi-classical approximation of the corresponding QM path inte-grals will give directly the desired result. On the contrary, the calculation of trace anomalies requires to control the full perturbative expansion of QM path integrals on curved spaces. The latter has been a favorite topic of study over the years. Most of the early literature dealt with ways of deriving discretized expressions, but the program of taking the con-tinuum limit till the end to identify the correct regularization scheme to be used directly in the continuum was almost never completed.1

Thus, starting from Refs.关3,4兴 a critical re-examination of the correct definitions of QM path integrals was initiated which lead to two well-defined and consistent schemes of regulating and computing: mode regularization共MR兲 关3,4,7兴 and time slicing 共TS兲 关8,7兴. Recently, following the sugges-tion in Ref. 关9兴 of using dimensional regularization a third

way of properly defining the path integrals has been devel-oped in 关10兴: the dimensional regularization scheme 共DR兲. While the counterterms required in MR and TS are nonco-variant, they happen to be covariant in the DR scheme 关10,11,9兴.

It is the purpose of this paper to describe in details this new scheme, test further its consistency and show the tech-nical advantage of having to deal with a manifestly covariant action in performing higher loop calculations for 0⫹1 non-linear sigma models on a finite time interval 共i.e., quantum mechanics on curved spaces with a finite propagation time兲. We apply the DR regulated path integral to compute the trace anomaly of a conformal scalar in six dimensions共for a review on trace anomalies see 关12兴兲. The correct full trace anomaly for such a scalar 共and also other six dimensional conformal free fields兲 has only recently been calculated in 关13兴 by using the heat kernel results of Gilkey 关14兴. With a DR path integral calculation we are going to reproduce the complete expression of this anomaly.

The paper is structured as follows. In Sec. II we review the DR scheme, in Sec. III we use it to calculate the trace anomaly for a conformal scalar field in 6D and in Sec. IV we present our conclusions. Finally, in Appendix A we report a list of structures and integrals employed in the main text: since the complete calculation is somewhat lengthy, it is use-ful for comparison purposes and future reference to record intermediate results.

II. DIMENSIONAL REGULARIZATION OF THE PATH INTEGRAL

First, we briefly review the quantization with path inte-grals of the motion of a共nonrelativistic and unit mass兲 par-ticle on a curved space with metric gi j(x) and scalar

poten-tial V(x). The model is described by the Euclidean action2 *Email address: bastianelli@bo.infn.it

†_{Email address: olindo@insti.physics.sunysb.edu}

1_{One exception is the description of the phase-space path integral}

in the book of Sakita关6兴: it contains noncovariant counterterms and the Feynman rules given there can be used to compute to any de-sired loop order since no ambiguous product of distributions is ever to be found in the loop expansion关7兴. The same cannot be said for the configuration space version: ambiguities are present there and must be resolved with a consistent scheme for multiplying

distribu-tions, which is what a regularization scheme provides. 2We perform a Wick rotation on the time variable and work con-0556-2821/2001/63共6兲/065005共11兲/$15.00 63 065005-1 ©2001 The American Physical Society

S关xi兴⫽

冕

t_i t_f dt

冋

1 2gi j共x兲x˙ i_x˙j_⫹V共x兲

册

_{. 共1兲}

In canonical quantization one must choose an ordering con-sistent with reparametrization invariance to produce a quan-tum Hamiltonian H⫽⫺12ⵜ

2_{⫹␣R⫹V 共our curvature}

con-ventions are found in Appendix A兲. The value of the parameter␣ depends on the particular order chosen关15兴 and conventionally can be taken to vanish with the agreement of reintroducing the coupling to R through the potential V.

Using path integrals the canonical ordering ambiguities re-emerge as the need of specifying a regularization scheme. The 1D sigma model in Eq. 共1兲 contains double derivative interactions which make Feynman graphs superficially diver-gent at one and two loops. However, the nontrivial path in-tegral measure can be exponentiated using ghost fields: their effect is to make finite the sum of the Feynman graphs but a regularization scheme is still necessary to render finite each individual divergent graph. Different regularization schemes require different counterterms to reproduce a quantum Hamiltonian with ␣⫽0. In mode regularization and time slicing such counterterms are noncovariant. In the DR scheme the counterterm is covariant and equal to VDR

⫽R/8, as demonstrated in 关10,11兴.

Now, let us describe the DR scheme which we are going to apply in the next section. First of all, we find it convenient to use a rescaled time parameter␶by defining t⫽␤␶⫹t_fand ␤⫽tf⫺ti, so that␶ will take values on the finite interval I

⬅关⫺1,0兴. Then, we introduce bosonic ai _{and fermionic}

bi,ci ghosts to exponentiate the nontrivial part of the path integral measure: integrating them back will formally repro-duce the 兿

冑

detg_{i j} factor of the measure. Finally, we intro-duce D extra infinite regulating dimensions t⫽(t1, . . . ,tD) with the prescription that one will take the limit D→0 at the very end of all calculations. Denoting t␮⬅(␶,t) with ␮ ⫽0,1, . . . ,D and dD⫹1_t⫽d␶dD_{t, the action in D⫹1}

dimen-sions reads S关x,a,b,c兴⫽1 ␤

冕

⍀d D⫹1_t

冋

1 2gi j共x兲共⳵␮x i_⳵ ␮xj⫹aiaj⫹bicj兲 ⫹␤2_{V共x兲⫹␤}2_V DR共x兲

册

共2兲

where VDR⫽R/8 is the counterterm in dimensional

regular-ization and⍀⫽I⫻RDis the region of integration containing the finite interval I.

The perturbative expansion can be generated by first de-composing the paths xi(␶) into a classical part x_cli (␶) satis-fying the boundary conditions and quantum fluctuations qi(␶) which vanish at the boundary 共the ghost fields are taken to vanish at the boundary as well兲 and then decompos-ing the Lagrangian into a free part plus interactions. The latter step is achieved by Taylor expanding the metric and

the potential around a fixed point, which we choose to be the final point xf. Thus, the propagators are recognized to be

具

xi共t兲xj共s兲典⫽⫺␤ gi j共xf兲 ⌬共t,s兲

具

ai共t兲aj共s兲典⫽␤ gi j共xf兲 ⌬gh共t,s兲,

具

bi共t兲cj共s兲典⫽⫺2␤ gi j共xf兲 ⌬gh共t,s兲 共3兲 with ⌬共t,s兲⫽

冕

d D_k 共2␲兲D ⫻

兺

n⫽1 ⬁ _⫺2

共␲n兲2⫹k2sin共␲n␶兲sin共␲n␴兲e

ik•(t⫺s) 共4兲 ⌬gh共t,s兲⫽

冕

dDk 共2␲兲D_n

兺

_⫽1 ⬁

2 sin共␲n␶兲sin共␲n␴兲eik•(t⫺s) ⫽␦D⫹1_{共t,s兲⫽␦共␶,␴兲␦}D_{共t⫺s兲 共5兲} where ␦共␶,␴兲⫽

兺

n⫽1 ⬁ 2 sin共␲n␶兲sin 共␲n␴兲 共6兲

is the Dirac delta on the space of functions vanishing at ␶,␴⫽⫺1,0. Of course, the function ⌬(t,s) satisfies the Green equation

⳵␮2⌬共t,s兲⫽⌬gh共t,s兲⫽␦D⫹1共s,t兲. 共7兲

The D→0 limits of these propagators are the usual ones ⌬共␶,␴兲⫽␶共␴⫹1兲␪共␶⫺␴兲⫹␴共␶⫹1兲␪共␴⫺␶兲 共8兲 ⌬gh共␶,␴兲⫽••⌬共␶,␴兲⫽␦共␶,␴兲 共9兲

where dots on the left or right side denote derivatives with respect to the first or second variable, respectively. However, such limits can be used only after one has cast the integrands corresponding to the various Feynman diagrams in an unam-biguous from by making use of the manipulations allowed by the regularization scheme. In particular, in DR one can use partial integration: it is always allowed in the extra D dimen-sion because of momentum conservation, while it can be performed along the finite time interval whenever there is an explicit function which vanishes at the boundary 关for ex-ample the propagator of the coordinates ⌬(t,s)]. Along the way one may find terms of the form⳵_␮2⌬(t,s) which accord-ing to Eq.共7兲 gives Dirac delta functions. The latter can be safely used at the regulated level, i.e., in D⫹1 dimensions. By performing such partial integrations one tries to arrive at an unambiguous form of the integrals which can be safely and easily calculated even after the limit D→0 is taken.

An explicit example will suffice to describe how the above rules are concretely used:

冕

⫺1 0 d␶

冕

⫺1 0 d␴ 共•_{⌬兲 共⌬}•_{兲 共}•_⌬•_兲 →

冕

dD⫹1t

冕

dD⫹1s 共_␮⌬兲 共⌬_␯兲 共_␮⌬_␯兲 ⫽

冕

dD⫹1t

冕

dD⫹1s 共_␮⌬兲 _␮

冉

1 2共⌬␯兲 2

冊

⫽⫺1₂

冕

dD⫹1t

冕

dD⫹1s 共_␮␮⌬兲 共⌬_␯兲2 ⫽⫺1₂

冕

dD⫹1t

冕

dD⫹1s ␦D⫹1共t,s兲 共⌬_␯兲2 ⫽⫺1₂

冕

dD⫹1t 共⌬_␯兲2兩t→⫺ 1 2

冕

_⫺1 0 d␶ 共⌬•兲2兩_␶ ⫽⫺₂₄1

where the symbol兩_␶means that one should set␴⫽␶. Thus, we see that the rules of computing in DR are quite similar to those used in MR, the only diversity being in the different options allowed in partial integrations. In DR the rule for contracting which indices with which indices follows directly from the regulated action in Eq.共2兲 and only certain partial integrations are allowed in D⫹1 dimensions. In MR one regulates by cutting off all mode sums at a large mode N and then performs partial integrations: all derivatives are now of the same nature and different options of partial inte-grations arise. This explains the origin of different counter-terms for these two regularizations.

III. TRACE ANOMALIES FOR A CONFORMAL SCALAR IN 6D

As described in 关3,4兴, one-loop trace anomalies can be obtained by computing a certain Fujikawa Jacobian suitably regulated and represented as a quantum mechanical path in-tegral with periodic boundary conditions

冕

d6_x

冑

_{g ␴共x兲}

_具

_Ta a共x兲

典

⫽ lim ␤→0 Tr关␴ e⫺␤H兴 ⫽ lim ␤→0

冕

PBC Dx ␴共x兲 e⫺S[x]_, 共10兲 where on the left hand side Taadenotes the trace of the stress

tensor for a 6D conformal scalar and ␴(x) is an arbitrary function describing an infinitesimal Weyl variation. In the first equality the infinitesimal part of the Fujikawa Jacobian has been regulated with the conformal scalar field kinetic operator H⫽⫺12ⵜ

2_⫺ 1

10R. The limit␤→0 should be taken after removing divergent terms in ␤ 共which is what the renormalization of the scalar field QFT will do兲, and so it picks up just the ␤ independent term. Finally on the right hand side the trace is given a representation as a path integral corresponding to a model with Hamiltonian H and with pe-riodic boundary conditions. The latter can be obtained using the quantum mechanics described in the previous section with a scalar potential V⫽⫺ 1

10R.

Thus we start computing the terms in the loop expansion of the path integral described in Sec. II. It will soon be clear that it is enough to compute up to order ␤3, i.e., up to 4 loops 关␤ can be taken as the loop counting parameter, as evident form Eq. 共2兲兴. We use reparametrization invariance and choose Riemann normal coordinates centered at the point x0

i

representing the boundary conditions at ␶⫽⫺1,0, and which will be integrated over to recreate the full periodic boundary conditions on the right hand side of Eq. 共10兲.

The expansion of the metric in Riemann normal coordi-nates is well known. For our case, since the action including the counterterm is manifestly covariant, that expansion can be easily generated by the method described for this context in 关3兴. One obtains the following terms needed in our ap-proximation: gmn共x兲dxmdxn⫽

冋

gmn⫹ 1 3Rmabnx a_xb_⫹ 1 3!ⵜiRmabnx a_xb_xi_⫹ 6 5!

冉

ⵜiⵜjRmabn⫹ 8 9Rmab ␣_R ␣i jn

冊

xaxbxixj ⫹_5!1 4₃共ⵜiⵜjⵜkRmabn⫹4Rmab␣ⵜiR␣ jkn兲xaxbxixjxk⫹ 10 7!

冉

ⵜiⵜjⵜkⵜlRmabn⫹ 34 5 Rmi j ␣_ⵜ kⵜlR␣abn ⫹11₂ ⵜiRmab␣ⵜjR␣kln⫹ 8 5Rmab ␣_R ␣i j␤R␤kln

冊

xaxbxixjxkxl⫹•••

册

dxmdxn V共x兲⫽V⫹共ⵜiV兲xi⫹ 1 2共ⵜiⵜjV兲x i_xj_⫹ 1 3!共ⵜiⵜjⵜkV兲x i_xj_xk_⫹ 1 4!共ⵜiⵜjⵜkⵜlV兲x i_xj_xk_xl_{⫹••• 共11兲}

where all tensorial quantities on the right hand sides are evaluated at the origin of the coordinate system. Notice that for the MR and TS regularization schemes the counterterms are noncovariant and their expansions cannot be generated so easily: obtaining the vertices from those counterterms would require a tedious computation.

Plugging the above expansions in the action共2兲 and noticing that the factor␤2raises by two the loop order for each vertex coming from the potential or the counterterm, we compute3

具

x0,␤兩x0,0

典⫽

冕

Dxe⫺S⫽A

具

e⫺Sint

典

⫽A exp

冋

⫺

具

S4

典⫺

具

S6

典⫺

具

S8

典⫹

1 2

具

S4 2

_典

c⫹ 1 2

具

S5 2

_典

c⫹具S4S6

典

c⫺ 1 6

具

S4 3

_典

c⫹O共␤4兲

册

共12兲

where the subscript ‘‘c’’ stands for connected diagrams only and where A⫽(2␲␤)⫺D/2gives the correct normalization of the path integral measure. Because of this normalization we see that for D⫽6 the␤-independent term is obtained by picking up the ␤3 contributions from the expansion of the exponential on the right hand side of Eq.共12兲.

The terms up to 3 loops are easily computed in DR by using the detailed expressions reported in 关16兴: one just needs to compute the integrals reported there using the DR rules. Including for simplicity the counterterm inside the potential V, we obtain

具

S4

典

⫽⫺␤

冋

1 24R⫺V

册

共13兲

具

S6

典

⫽⫺ ␤2 12

冋

1 40ⵜ 2_R⫹ 1 90Rmn 2 _⫹ 1 60Rmnab 2 _⫺ⵜ2_V

册

_共14兲

具

S₄2

典

_c⫽⫺␤ 2 72

冋

1 3Rmn 2

册

_{. 共15兲}

To achieve notational simplicity in the remaining 4-loop terms we use the basis of curvature invariants given in Appendix A and compute the terms reported there. We obtain

具

S8

典

⫽⫺ ␤3 7!

冋

17 15K4⫺ 16 15K5⫹ 8 5K6⫹ 5 12K7⫺ 8 3K8 ⫹11₁₀K10⫹ 3 2K11⫺ 19 20K12⫹ 149 48 K13⫺ 25 8 K14⫹ 11 16K15⫺ 5 24K16⫹ 3 8K17

册

⫺␤ 3 6!关2R mn_ⵜ mⵜnV⫹ⵜmRⵜmV⫺3ⵜ4V兴 共16兲

具

S₅2

典

c⫽⫺ ␤3 6!

冋

23 24K13⫺ 3 4K14⫺ 1 8K15⫺ 5 48K16⫹5ⵜ m_Rⵜ mV⫺60共ⵜmV兲2

册

共17兲

具

S4S6

典

c⫽⫺ ␤3 6!

冋

13 45K4⫺ 1 5K5⫹ 2 15K6⫹ 3 10K10⫺ 1 10K12

册

共18兲

具

S4 3

_典

c⫽⫺ ␤3 6!

冋

2 3K4⫹ 1 6K7⫹ 4 3K8

册

. 共19兲

Inserting all these values into Eq.共12兲 we get

3_{Since x}i

(⫺1)⫽xi(0)⬅x0 the classical field is xcl(␶)⫽0; hence all diagrams with external fields vanish. We indicate with Sn the

interaction terms containing n fields when originating from the expansion of the metric and n⫺4 fields when originating from the scalar potential.

具

x0,␤兩x0,0

典

⫽ 1 共2␲␤兲3exp

冋

␤

冉

1 24R⫺V

冊

⫹␤ 2

冉

1 720共Rmnab 2 _⫺R mn 2 _兲⫹ 1 480ⵜ 2_R⫺ 1 12ⵜ 2_V

冊

⫹␤ 3 8!

冉

⫺ 8 9K4⫹ 8 3K5⫹ 16 3 K6⫹ 44 9 K7⫺ 80 9 K8⫺8K10⫹12K11⫺2K12 ⫺2K13⫺4K14⫹9K15⫹ 5 4K16⫹3K17

冊

⫹␤ 3 6!

冉

2R mn_ⵜ mⵜnV⫺ 3 2ⵜ m_Rⵜ mV⫹30共ⵜmV兲2⫺3ⵜ4V

冊

⫹O共␤4兲

册

. 共20兲

Now, using the value V⫽1 8R⫺

1 10R⫽

1

40R to take into ac-count the ac-counterterm VDRand the conformal coupling of the

scalar field, we compare with Eq.共10兲 and obtain the corre-sponding trace anomaly

具

Taa

典

⫽ 1 共2␲兲3 1 8!

冋

7 225K1⫺ 14 15K2⫹ 14 15K3⫺ 8 9K4⫹ 8 3K5 ⫹16₃ K6⫹ 44 9 K7⫺ 80 9 K8⫺8K10⫹12K11⫹ 4 5K12 ⫺2K13⫺4K14⫹9K15⫹ 1 5K16⫺ 6 5K17

册

. 共21兲 It agrees with the one given in关13兴, where it was shown that it can be cast also as

具

Taa

典⫽

1 共2␲兲3 1 8!

冋

⫺ 5 72E6⫺ 28 3 I1⫹ 5 3I2⫹2I3 ⫹trivial anomalies

册

共22兲 with the topological Euler density given by

E6⫽⫺⑀m₁n₁m₂n₂m₃n₃⑀a1b1a2b2a3b3Rm1n1a₁b₁

⫻Rm₂n₂

a₂b₂Rm3n3a₃b₃ 共23兲

and the three Weyl invariants

I1⫽CamnbCmi jnCiabj, 共24兲 I2⫽CabmnCmni jCi jab, 共25兲 I3⫽Cmabc

冉

ⵜ2␦n m_⫹4R n m_⫺6 5R␦n m

冊

Cnabc⫹trivial anomalies, 共26兲 whereas the coefficients of the trivial anomalies are unimpor-tant since they can be changed by the variation of local coun-terterms. The structure of trivial anomalies has been fully analyzed in关17兴. It is interesting to note, after inspecting the results in关17兴, that the coefficients of K1, K2and K3never

appear in the trivial anomalies. At the same time they are produced in the previous calculation by disconnected dia-grams. Thus one may fix three of the four true anomalies by a simpler lower loop calculation, while the remaining inde-pendent fourth nontrivial anomaly, which can be taken as the one corresponding to E₆, could be fixed by an independent calculation on the simplified geometry of a maximally sym-metric space.

IV. CONCLUSIONS

We have used the recently developed dimensional regu-larization scheme for quantum mechanical path integrals关10兴 to compute the trace anomaly for a scalar field in six dimen-sions. The identification of the full anomaly required a com-plete 4-loop quantum mechanical computation. Technically, the covariance of the counterterm VDR allows a more

effi-cient identification of the corresponding vertices than in the MR and TS regularization schemes.

We noticed that the coefficients of three of the four non-trivial anomalies could as well be obtained by a simpler 3-loop calculation. One may speculate that such a fact may happen also for D⫽8 trace anomalies: there one would need to compute the quantum mechanics up to 5-loops, but it could happen that all nontrivial anomalies but one could be fixed by a simpler 4-loop calculation 共presented in this pa-per兲 and the remaining one by a calculation on a simplified geometry. To concretely check this conjecture, one would need a cohomological analysis to identify the structure of all trivial and nontrivial anomalies, as the one given in Ref.关17兴 for the six dimensional case. However, such an analysis is not available in the literature yet.

One could couple the nonlinear sigma model to non-Abelian gauge potentials to obtain the trace anomalies of other six dimensional conformal fields 关13兴. In such an ex-tension the main new complication is related to the time ordering prescription to be used for achieving gauge covari-ance, as employed in 关4兴, which forces to compute different DR integrals for different ordering of the vertices. An ap-proach which could guarantee non-Abelian covariance in a more straightforward manner would clearly be welcome. It may be related to the extra ghost fields used in 关8兴.

viewing it as the calculation of a certain Fujikawa Jacobian, conceptually it can be thought of as performed in the first quantized approach of the scalar particle theory共see the discussion in 关19兴兲. Given that interpretation, it would be interesting to investigate if such worldline path integrals in curved space could be useful to simplify computations of scattering amplitudes and effective actions of perturbative QFT coupled to gravity, as it happens in the flat space case 关20兴.

APPENDIX A

We use the convention 关ⵜa,ⵜb兴Vc⫽RabcdVd, Rab⫽Raccb. It is useful for notational purposes to introduce a basis of

curvature invariants cubic in the curvature

K₁⫽R3 K₂⫽RR_ab2 K₃⫽RR_abmn2 K4⫽RamRmiRia K5⫽RabRmnRmabn K6⫽RabRamnlRbmnl K7⫽RabmnRmni jRi jab K8⫽RamnbRmi jnRiabj K9⫽Rⵜ2R K10⫽Rabⵜ2Rab K11⫽Rabmnⵜ2Rabmn K12⫽RabⵜaⵜbR K13⫽共ⵜaRmn兲2 K14⫽ⵜaRbmⵜbRam K15⫽共ⵜiRabmn兲2 K16⫽共ⵜaR兲2 K17⫽ⵜ4R. 共A1兲

It differs from the basis used in 关17,18兴 only in the definition of K16: the one used above enters more naturally in our

calculations.

In the main text contributions of order␤3 _{to the effective action come from the terms listed below. In the list of integrals}

we use the following conventions. The limits of integration are关⫺1,0兴 for all variables. For 3-dimensional integrals the first group of propagator in round brackets depends on (␶,␴), the second on (␴,␳) and the third on (␳,␶), with this precise order, while terms at coinciding points are explicitly indicated. For 2-dimensional integrals the propagators at non-coinciding points depend on (␶,␴), while for 1-dimensional integrals all terms are obviously taken at coinciding points. We use the shorthand notation •⌬•⫽•⌬•⫹••⌬. The DR regularization is immediate and we quote the DR values.

具

S₄3

典

c

具

S₄3

典

c⫽A0⫹A1⫹A2⫹A3 共A2兲

A0⫽ ␤3 9

冋冉

1 4K7⫹2K8

冊冉

I1 A₀ ⫺I2 A₀ ⫺I3 A₀ ⫹2I4 A₀ ⫹2I5 A₀ ⫺2I6 A₀ ⫹1₃I₁₁A0

冊

⫺

冉

7 2K7⫹K8

冊冉

1 3I7 A₀ ⫹I₉A₀

冊

⫹

冉

13 4 K7⫺K8

冊冉

I8 A₀ ⫹1 3I10 A₀

冊册

共A3兲 A1⫽ ␤3 6 K6共I1 A₁ ⫺I2 A₁ ⫺2I3 A₁ ⫹2I4 A₁ ⫹I5 A₁ ⫺I6 A₁ ⫹I7 A₁ ⫺I8 A₁ ⫺2I9 A₁ ⫹2I10 A₁ ⫹I₁₁A₁ ⫺I₁₂A₁ ⫺2I₁₃A₁ ⫹2I₁₄A₁ ⫹2I₁₅A₁ ⫺2I₁₆A₁ ⫹2I₁₇A₁ ⫺2I₁₈A₁ ⫺2I₁₉A₁ ⫹2I₂₀A₁ 兲 共A4兲 A2⫽ ␤3 9 K5关I1 A₂ ⫺I2 A₂ ⫹I3 A₂ ⫺I4 A₂ ⫹4I5 A₂ ⫺I6 A₂ ⫺2I7 A₂ ⫺I8 A₂ ⫹I9 A₂ ⫹I10 A₂ ⫹2共⫺I11 A₂ ⫺I12 A₂ ⫺I13 A₂ ⫹I14 A₂ ⫹I15 A₂ ⫺I16 A₂ ⫺I17 A₂ ⫹I18 A₂ ⫹I19 A₂ 兲兴 共A5兲

A3⫽ ␤3 27K4共I1 A₃ ⫺6I₂A₃ ⫹3I₃A₃ ⫹3I₄A₃ ⫺6I₅A₃ ⫺6I₆A₃ ⫹3I₇A₃ ⫹3I₈A₃ ⫹6I₉A₃ ⫹I₁₀A₃ ⫺6I₁₁A₃ ⫹3I₁₂A₃ ⫹3I₁₃A₃ ⫹6I₁₄A₃ ⫺2I₁₅A₃ ⫺6I₁₆A₃ 兲. 共A6兲 Integrals in A0: I₁A0⫽

冕 冕 冕

共⌬• 2_兲共⌬2_兲共•_⌬• 2_⫺••_⌬2_兲⫽⫺ 13 3780 I2 A₀ ⫽

冕 冕 冕

共⌬ ⌬•_兲共•_{⌬ ⌬兲共}•_⌬• 2_⫺••_⌬2_兲⫽⫺ 1 15120 I₃A0⫽

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共•_⌬• _⌬•_兲共⌬2_兲共•_⌬• •_⌬兲⫽ 13 7560 I4 A₀ ⫽

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共•_⌬• _⌬兲共•_{⌬ ⌬兲共}•_⌬• •_⌬兲⫽⫺ 13 15120 I₅A0⫽

冕 冕 冕

_共•_{⌬ ⌬}•_兲共•_{⌬ ⌬兲共}•_⌬• •_⌬兲⫽⫺ 1 3780 I6 A₀ ⫽

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共•_{⌬ ⌬兲共}•_⌬2_兲共•_⌬• •_⌬兲⫽ 1 1890 I₇A0⫽

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_关共⌬ •_⌬•_兲共⌬ •_⌬•_兲共⌬ •_⌬•_兲⫹共⌬ ••_⌬兲共⌬ ••_⌬兲共⌬ ••_{⌬兲兴⫽⫺} 43 15120 I₈A0⫽

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_共•_{⌬ ⌬}•_兲共⌬ •_⌬•_兲共⌬ •_⌬•_兲⫽ 11 15120 I9 A₀ ⫽

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共•_{⌬ ⌬}•_兲共•_{⌬ ⌬}•_兲共⌬ •_⌬•_兲⫽ 1 756 I₁₀A0⫽

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_共•_{⌬ ⌬}•_兲共•_{⌬ ⌬}•_兲共•_{⌬ ⌬}•_兲⫽⫺ 1 945 I11 A₀ ⫽

冕 冕 冕

共•_⌬2_兲共•_⌬2_兲共•_⌬2_兲⫽ 1 756. 共A7兲 Integrals in A1: I₁A1⫽

冕 冕 冕

_共•_⌬•_兲兩 ␶共⌬•兲共•⌬• ⌬2兲共•⌬兲⫽⫺ 13 3780 I2 A₁ ⫽

冕 冕 冕

共•_⌬•_兲兩 ␶共⌬•兲共•⌬ ⌬ ⌬•兲共•⌬兲⫽ 1 1890 I₃A1⫽

冕 冕 冕

_共•_⌬•_{兲兩␶共⌬}•_兲共•_⌬• _{⌬ ⌬}•_{兲共⌬兲⫽⫺} 1 945 I4 A₁ ⫽

冕 冕 冕

共•_⌬•_{兲兩␶共⌬}•_兲共•_{⌬ ⌬}• 2_{兲共⌬兲⫽⫺} 1 3780 I₅A1⫽

冕 冕 冕

共•_⌬•_兲兩 ␶共⌬兲共⌬关•⌬• 2⫺••⌬2兴兲共⌬兲⫽ 1 432 I6 A₁ ⫽

冕 冕 冕

共•_⌬•_兲兩 ␶共⌬兲共•⌬• ⌬• •⌬兲共⌬兲⫽⫺ 1 15120 I₇A1_⫽

冕 冕 冕

_共⌬兲兩 ␶关共•⌬•兲共⌬2 •⌬•兲共•⌬•兲⫹共••⌬兲共⌬2 ••⌬兲共••⌬兲兴⫽⫺ 1 216 I₈A1⫽

冕 冕 冕

_共⌬兲兩 ␶共•⌬•兲共•⌬ ⌬ ⌬•兲共•⌬•兲⫽⫺ 1 15120 I₉A1⫽

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_共⌬兲兩 ␶共•⌬•兲共•⌬• ⌬ ⌬•兲共⌬•兲⫽⫺ 1 2160 I10 A₁ ⫽

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共⌬兲兩␶共•⌬•兲共•⌬ ⌬• 2兲共⌬•兲⫽ 2 945 I₁₁A1⫽

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_{共⌬兲兩␶共}•_{⌬兲共⌬关}•_⌬• 2_⫺••_⌬2_兴兲共⌬•_兲⫽⫺ 19 15120 I12 A₁ ⫽

冕 冕 冕

共⌬兲兩␶共•_⌬兲共•_⌬• •_{⌬ ⌬}•_兲共⌬•_兲⫽⫺ 1 1512 I₁₃A1⫽

冕 冕 冕

共•_{⌬兲兩␶共}•_⌬•_兲共•_⌬• _⌬2_兲共•_⌬兲⫽ 13 7560 I14 A₁ ⫽

冕 冕 冕

共•_{⌬兲兩␶共}•_⌬•_兲共•_{⌬ ⌬ ⌬}•_兲共•_⌬兲⫽⫺ 1 3780 I₁₅A1_⫽

冕 冕 冕

_共•_{⌬兲兩␶共}•_⌬•_兲共•_⌬• _⌬• _{⌬兲共⌬兲⫽} 17 15120 I16 A1_⫽

冕 冕 冕

_共•_{⌬兲兩␶共}•_⌬•_兲共•_{⌬ ⌬}• 2_{兲共⌬兲⫽} 1 7560 I₁₇A1⫽

冕 冕 冕

_共•_⌬兲兩 ␶共•⌬兲共•⌬• •⌬ ⌬兲共•⌬兲⫽⫺ 1 15120 I18 A1⫽

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_共•_⌬兲兩 ␶共•⌬兲共•⌬2 ⌬•兲共•⌬兲⫽ 1 7560 I₁₉A1⫽

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_共•_⌬兲兩 ␶共•⌬兲共⌬关•⌬• 2⫺••⌬2兴兲共⌬兲⫽⫺ 1 864 I20 A₁ ⫽

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共•_⌬兲兩 ␶共•⌬兲共•⌬• •⌬ ⌬•兲共⌬兲⫽ 1 30240. 共A8兲

Integrals in A2: I₁A2⫽

冕 冕 冕

共•_⌬•_兲兩 ␶共•⌬•兲兩␴共1兲共⌬2兲共•⌬2兲⫽ 1 756 I2 A₂ ⫽

冕 冕 冕

共•_⌬•_兲兩 ␶共•⌬•兲兩␴共1兲共⌬ ⌬•兲共•⌬ ⌬兲⫽ 1 1890 I₃A2⫽

冕 冕 冕

共⌬兲兩 ␶共⌬兲兩␴共1兲共•⌬2兲共•⌬• 2⫺••⌬2兲⫽⫺ 11 1890 I4 A₂ ⫽

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共⌬兲兩␶共⌬兲兩␴共1兲共•⌬ •⌬•兲共•⌬• ⌬•兲⫽ 2 945 I₅A2_⫽

冕 冕 冕

_共⌬•_兲兩 ␶共⌬•兲兩␴共1兲共•⌬ ⌬兲共•⌬• •⌬兲⫽ 1 3024 I6 A2_⫽

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_共⌬•_兲兩 ␶共⌬•兲兩␴共1兲共•⌬ ⌬•兲共•⌬ ⌬•兲⫽ 1 30240 I₇A2⫽

冕 冕 冕

_共⌬•_兲兩 ␶共⌬•兲兩␴共1兲共⌬ •⌬•兲共•⌬ ⌬•兲⫽⫺ 1 6048 I8 A2⫽

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_共⌬•_兲兩 ␶共⌬•兲兩␴共1兲共⌬ •⌬•兲共⌬ •⌬•兲⫽ 5 6048 I₉A2⫽

冕 冕 冕

_共•_⌬•_兲兩 ␶共⌬兲兩␴共1兲共•⌬2兲共•⌬2兲⫽⫺ 1 378 I10 A₂ ⫽

冕 冕 冕

共•_⌬•_兲兩 ␶共⌬兲兩␴共1兲共•⌬• 2⫺••⌬2兲共⌬2兲⫽ 11 3780 I₁₁A2⫽

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_共•_⌬•_{兲兩␶共⌬兲兩␴共1兲共}•_⌬ •_⌬•_兲共•_{⌬ ⌬兲⫽⫺} 1 945 I12 A₂ ⫽

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共•_⌬•_{兲兩␶共⌬}•_{兲兩␴共1兲共}•_⌬⌬兲共•_⌬2_兲⫽⫺ 1 1512 I₁₃A2⫽

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共•_⌬•_兲兩 ␶共⌬•兲兩␴共1兲共•⌬• ⌬•兲共⌬2兲⫽⫺ 1 1512 I14 A₂ ⫽

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共•_⌬•_兲兩 ␶共⌬•兲兩␴共1兲共⌬ •⌬•兲共•⌬ ⌬兲⫽⫺ 1 1512 I₁₅A2_⫽

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_共•_⌬•_兲兩 ␶共⌬•兲兩␴共1兲共•⌬ ⌬•兲共•⌬ ⌬兲⫽ 1 7560 I16 A2_⫽

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_共⌬兲兩 ␶共⌬•兲兩␴共1兲共•⌬⌬兲共•⌬• 2⫺••⌬2兲⫽⫺ 11 7560 I₁₇A2⫽

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_共⌬兲兩 ␶共⌬•兲兩␴共1兲共•⌬• ⌬•兲共⌬• 2兲⫽ 1 756 I18 A2⫽

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_共⌬兲兩 ␶共⌬•兲兩␴共1兲共⌬ •⌬•兲共•⌬• ⌬•兲⫽ 1 756 I₁₉A2⫽

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_共⌬兲兩 ␶共⌬•兲兩␴共1兲共•⌬ ⌬•兲共•⌬• ⌬•兲⫽⫺ 1 3780. 共A9兲 Integrals in A₃: I₁A3_⫽

冕 冕 冕

_共•_⌬•_兲兩 ␶共•⌬•兲兩␴共•⌬•兲兩␳共⌬兲共⌬兲共⌬兲⫽⫺ 1 945 I2 A3_⫽

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_共•_⌬•_兲兩 ␶共•⌬•兲兩␴共•⌬兲兩␳共⌬兲共⌬•兲共⌬兲⫽ 1 1890 I₃A3⫽

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_共•_⌬•_兲兩 ␶共•⌬•兲兩␴共⌬兲兩␳共⌬兲共⌬•兲共•⌬兲⫽ 1 756 I4 A3⫽

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_共•_⌬•_兲兩 ␶共⌬兲兩␴共⌬兲兩␳共⌬•兲共•⌬•兲共•⌬兲⫽⫺ 29 7560 I₅A3⫽

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_共•_⌬•_兲兩 ␶共⌬•兲兩␴共⌬兲兩␳共⌬兲共•⌬•兲共•⌬兲⫽⫺ 13 15120 I6 A₃ ⫽

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共•_⌬•_兲兩 ␶共⌬•兲兩␴共⌬兲兩␳共⌬•兲共⌬•兲共•⌬兲⫽⫺ 1 2160 I₇A3⫽

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_共•_⌬•_{兲兩␶共⌬}•_{兲兩␴共⌬}•_{兲兩␳共⌬}•_{兲共⌬兲共}•_⌬兲⫽⫺ 11 30240 I8 A₃ ⫽

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共•_⌬•_{兲兩␶共⌬}•_{兲兩␴共⌬}•_{兲兩␳共⌬兲共}•_⌬•_{兲共⌬兲⫽⫺} 11 30240 I₉A3⫽

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共•_⌬•_兲兩 ␶共⌬•兲兩␴共⌬•兲兩␳共⌬•兲共⌬•兲共⌬兲⫽⫺ 1 6048 I₁₀A3_⫽

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_共⌬兲兩 ␶共⌬兲兩␴共⌬兲兩␳关共•⌬•兲共•⌬•兲共•⌬•兲⫹共••⌬兲共••⌬兲共••⌬兲兴⫽⫺ 71 7560 I₁₁A3⫽

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_共⌬兲兩 ␶共•⌬兲兩␴共⌬兲兩␳共•⌬•兲共⌬•兲共•⌬•兲⫽ 29 15120 I12 A3⫽

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_共⌬兲兩 ␶共•⌬兲兩␴共•⌬兲兩␳共•⌬兲共•⌬•兲共⌬•兲⫽ 19 30240 I₁₃A3⫽

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_共⌬兲兩 ␶共•⌬兲兩␴共•⌬兲兩␳共•⌬•兲共⌬兲共•⌬•兲⫽ 31 30240 I14 A₃ ⫽

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共⌬兲兩␶共•⌬兲兩␴共•⌬兲兩␳共•⌬兲共•⌬兲共•⌬•兲⫽⫺ 1 6048 I₁₅A3⫽

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_共•_{⌬兲兩␶共}•_{⌬兲兩␴共}•_{⌬兲兩␳共}•_⌬兲共•_⌬兲共•_⌬兲⫽⫺ 1 60480 I16 A₃ ⫽

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共•_{⌬兲兩␶共}•_{⌬兲兩␴共}•_{⌬兲兩␳共⌬兲共}•_⌬兲共•_⌬•_兲⫽ 11 60480. 共A10兲

具

S5 2

_典

c

具

S₅2

典

c⫽⫺ ␤3 144关K15共6I1 B₀ ⫺12I2 B₀ ⫹6I3 B₀ 兲⫹K16共4I1 B₂ ⫺8I2 B₂ ⫹4I3 B₂ 兲⫹K13共2I1 B₁ ⫺8I2 B₁ ⫹4I3 B₁ ⫹22I4 B₁ ⫺20I5 B₁ ⫺48I6 B₁ ⫹40I7 B₁ ⫹24I₈B₁ ⫺16I₉B₁ 兲⫹K14共4I₁ B₁ ⫺16I₂B₁ ⫹8I₃B₁ ⫺20I₄B₁ ⫹24I₅B₁ ⫹32I₆B₁ ⫺48I₇B₁ ⫺16I₈B₁ ⫹32I₉B₁ 兲兴 ⫺␤ 3 3 关3共ⵜaV兲 2_I 1 B₃ ⫹ⵜkRⵜkV共I₂ B₃ ⫺I₃B₃ 兲兴. 共A11兲 Integrals: I₁B0⫽

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⌬3_共•_⌬• 2_⫺••_⌬2_{兲⫽0 I} 2 B₀ ⫽

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⌬2 •_⌬ •_⌬• _⌬•_⫽ 1 840 I₃B0⫽

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⌬共•_⌬兲2_共⌬•_兲2_⫽⫺ 1 560 I1 B₁ ⫽

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⌬3_⫽⫺ 1 560 I₂B1⫽

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•_⌬兩 ␴⌬2⌬•⫽ 1 1680 I3 B₁ ⫽

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⌬兩␴ ⌬ 共⌬•兲2⫽ 1 336 I₄B1⫽

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⌬兩 ␶ ⌬兩␴ ⌬共•⌬• 2⫺••⌬2兲⫽ 17 2520 I5 B₁ ⫽

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⌬兩␶ ⌬兩␴ •⌬ •⌬• ⌬•⫽⫺ 1 2520 I₆B1⫽

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⌬兩 ␶ •⌬兩␴ •⌬ •⌬• ⌬⫽⫺ 1 720 I7 B₁ ⫽

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⌬兩␶ •⌬兩␴ 共•⌬兲2⌬•⫽⫺ 1 5040 I₈B1⫽

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•_⌬兩 ␶ •⌬兩␴ ⌬2 •⌬•⫽⫺ 1 1008 I9 B₁ ⫽

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•_⌬兩 ␶ •⌬兩␴ •⌬⌬⌬•⫽ 1 5040 I₁B2⫽

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⌬兩 ␶ ⌬兩␴ ⌬⫽⫺ 17 5040 I2 B₂ ⫽

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⌬兩␶共•⌬兩␴兲2⌬⫽ 1 1260 I₃B2⫽

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_共•_⌬兩 ␶兲2共•⌬兩␴兲2⌬⫽⫺ 1 4032 I1 B₃ ⫽

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⌬⫽⫺₁₂1 I₂B3⫽

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⌬兩 ␶ ⌬⫽ 1 60 I3 B₃ ⫽

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共•_⌬兩 ␶兲2⌬⫽⫺ 1 240. 共A12兲

具

S4S6

典

c

具

S4S6

典

c⫽⫺ ␤3 5!

再

共4K6⫹2K7⫺8K8⫹5K11兲共I2 C₁ ⫹I3 C₁ ⫺2I4 C₁ 兲⫹共⫺2K4⫹2K5⫹K10⫹3K12兲共I1 C₂ ⫹I2 C₂ ⫺2I3 C₂ 兲 ⫹共2K4⫺2K5⫹3K10⫺K12兲共I₄ C₂ ⫹I₅C₂ ⫺2I₆C₂ 兲⫹共⫺4K4⫹4K5⫺6K10⫹2K12兲共I₇ C₂ ⫹I₈C₂ ⫺I₉C₂ ⫺I₁₀C₂ 兲 ⫹共4K4⫺4K5⫹2K10⫺4K12兲共I₁₁ C₂ ⫹I₁₂C₂ ⫺2I₁₃C₂ 兲⫹4₉关共3K6⫺7K7⫺2K8兲共I₂ C₁ ⫹I₃C₁ ⫺2I₄C₁ 兲 ⫹共4K5⫹6K6兲共I1 C₂ ⫹I₂C₂ ⫺2I₃C₂ 兲⫹共2K4⫹3K6兲共I4 C₂ ⫹I₅C₂ ⫺2I₆C₂ 兲⫹共⫺4K4⫺8K5⫺6K10兲共I7 C₂ ⫹I₈C₂ ⫺I₉C₂ ⫺I₁₀C₂ 兲 ⫹共2K4⫺4K5⫺3K6兲共I11 C₂ ⫹I12 C₂ ⫺2I13 C₂ 兲兴

冎

⫺␤ 3 6 R mn_ⵜ nⵜmV共I1 C₃ ⫹I2 C₃ ⫺2I3 C₃ 兲. 共A13兲

Integrals: I₁C1⫽

冕 冕

共•_⌬•_兲兩 ␶ ⌬2共⌬•兲2⫽ 1 420 I2 C₁ ⫽

冕 冕

⌬兩␶ 共•⌬兲2共⌬•兲2⫽⫺ 1 420 I₃C1⫽

冕 冕

⌬兩 ␶ ⌬2共•⌬• 2⫺••⌬2兲⫽ 1 280 I4 C₁ ⫽

冕 冕

⌬兩␶ ⌬ •⌬ •⌬• ⌬•⫽ 1 1680 I₅C1⫽

冕 冕

•_⌬兩 ␶ ⌬ •⌬共⌬•兲2⫽0 I6 C₁ ⫽

冕 冕

•_⌬兩 ␶ ⌬2 •⌬• ⌬•⫽⫺ 1 840 I₁C2⫽

冕 冕

_共⌬ •_⌬•_兲兩 ␶ 共•⌬•兲兩␴ ⌬2⫽⫺ 1 420 I2 C₂ ⫽

冕 冕

共⌬ •_⌬•_兲兩 ␶ ⌬兩␴ 共⌬•兲2⫽ 1 210 I₃C2⫽

冕 冕

_共⌬ •_⌬•_兲兩 ␶ •⌬兩␴ ⌬⌬•⫽ 1 840 I4 C₂ ⫽

冕 冕

共⌬2_兲兩 ␶ 共•⌬•兲兩␴ 共•⌬兲2⫽ 1 252 I₅C2⫽

冕 冕

_共⌬2_兲兩 ␶ ⌬兩␴ 共•⌬• 2⫺••⌬2兲⫽ 11 1260 I6 C₂ ⫽

冕 冕

共⌬2_兲兩 ␶ •⌬兩␴ •⌬ •⌬•⫽⫺ 1 504 I₇C2⫽

冕 冕

_共⌬ •_⌬兲兩 ␶ 共•⌬•兲兩␴•⌬⌬⫽ 1 1260 I8 C₂ ⫽

冕 冕

共⌬ •_⌬兲兩 ␶ ⌬兩␴ ⌬• •⌬•⫽⫺ 1 630 I₉C2⫽

冕 冕

共⌬ •_⌬兲兩 ␶ •⌬兩␴ ⌬ •⌬•⫽⫺ 1 1008 I10 C₂ ⫽

冕 冕

共⌬ •_⌬兲兩 ␶ •⌬兩␴ •⌬⌬•⫽ 1 5040 I₁₁C2⫽

冕 冕

共•_⌬2_兲兩 ␶ 共•⌬•兲兩␴ ⌬2⫽ 1 2520 I12 C₂ ⫽

冕 冕

共•_⌬2_兲兩 ␶ ⌬兩␴ 共⌬•兲2⫽⫺ 1 1260 I₁₃C2_⫽

冕 冕

_共•_⌬2_兲兩 ␶ •⌬兩␴ ⌬⌬•⫽⫺ 1 5040 I1 C₃ ⫽

冕 冕

共•_⌬•_兲兩 ␶ ⌬2⫽ 1 90 I₂C3⫽

冕 冕

⌬兩 ␶ •⌬2⫽⫺ 1 45 I3 C₃ ⫽

冕 冕

•⌬兩␶ •⌬⫽⫺ 1 180. 共A14兲

具

S₈

典

具

S₈

典

⫽␤ 3 7!共I1 D_⫺I 2 D_兲

冋

_4共2K 4⫹9K6⫺7K7⫺2K8兲⫹ 55 2共11K13⫺10K14⫹3K15兲⫹34共2K4⫺2K5⫹4K6⫹2K7⫺8K8⫹3K10 ⫹5K11⫺K12兲⫹5共12K4⫺12K5⫹4K6⫹2K7⫺8K8⫹6K10⫹2K11⫺16K12⫹14K13⫺20K14 ⫺5K16⫹9K17兲

册

⫺ ␤3 24I3 D_共2Rmn_ⵜ mⵜnV⫹ⵜmRⵜmV⫺3ⵜ4V兲. 共A15兲 Integrals: I₁D⫽

冕

⌬3共•⌬•⫹⌬••兲⫽⫺ 1 140; I2 D_⫽

冕

_⌬2_⌬•2_⫽ 1 840; I3 D_⫽

冕

_⌬2_⫽1 30. 共A16兲

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