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# The Super-Heat-Kernel Expansion and the Renormalization of the Pion-Nucleon Interaction

CERN-TH/98-231 UWThPh-1998-42

arXiv:hep-ph/9807425v1 20 Jul 1998

The Super-Heat-Kernel Expansion and the Renormalization of the Pion¨CNucleon Interaction*
H. Neufeld

CERN, CH-1211 Geneva 23, Switzerland and Institut f¡§ ur Theoretische Physik, Universit¡§ at Wien Boltzmanngasse 5, A-1090 Vienna, Austria

Abstract A recently proposed super-heat-kernel technique is applied to SU (2)L ¡Á SU (2)R heavy baryon chiral perturbation theory. A previous result for the one-loop divergences of the pion¨Cnucleon system to O(p3 ) is con?rmed, giving at the same time an impressive demonstration of the e?ciency of the new method. The cumbersome and tedious calculations of the conventional approach are now reduced to a few simple algebraic manipulations. The present computational scheme is not restricted to chiral perturbation theory, but can easily be applied or extended to any (in general non-renormalizable) theory with boson¨Cfermion interactions.

CERN-TH/98-231 July 1998 * Work supported in part by TMR, EC-Contract No. ERBFMRX-CT980169 (EURODA¦µNE)

1

Introduction

The modern treatment of the pion¨Cnucleon system as an e?ective ?eld theory of the ¡¦ standard model was pioneered by Gasser, Sainio, Svarc [1] and Krause [2] who formulated the ¡°relativistic¡± version of baryon chiral perturbation theory. It was then shown by Jenkins and Manohar [3] that the methods of heavy quark e?ective theory [4] allow for a systematic low-energy expansion of baryonic Green functions in complete analogy to the meson sector. The latter approach is usually called heavy baryon chiral perturbation theory. Applications of this e?ective ?eld theory beyond the tree level require the knowledge of the divergences generated by one-loop graphs. For the pion¨Cnucleon interaction in the heavy mass expansion, the full list of one-loop divergences to O(p3 ) has been worked out by Ecker [5]. This analysis was then extended to the three-?avour case by M¡§ uller and Mei?ner [6]. In these papers, the bosonic loop and the mixed loop (boson and fermion lines in the loop) were treated separately. This required a cumbersome investigation of the singular behaviour of products of propagators, because the mixed loop does not have the form of a determinant, like the purely bosonic or fermionic loops. To overcome these di?culties, we have recently developed [7] a method where bosons and fermions are treated on an equal footing. Employing the notions of supermatrices, superdeterminants and supertraces [8, 9], we have constructed a super-heat-kernel representation for the one-loop functional of a boson-fermion system. In this approach, the determination of the one-loop divergences is reduced to simple matrix manipulations, in complete analogy to the familiar heat-kernel expansion technique for bosonic or fermionic loops. The present paper is organized as follows: in Sect. 2 I brie?y review the super-heatkernel method. In contrast to the Euclidean space formulation used in [7], the presentation in this work refers to Minkowski space throughout. In Sect. 3 the super-heat-kernel formalism is applied to a rather general class of scalar¨Cheavy fermion interactions (including, of course, heavy baryon chiral perturbation theory). The one-loop divergences to second order in the fermion ?elds are given explicitly. These results are then specialized to the two-?avour version of heavy baryon chiral perturbation theory in Sect. 4. My conclusions, together with an outlook to possible extensions of the present work, are summarized in Sect. 5. Several momentum-space integrals are collected in the Appendix.

2

Super-Heat-Kernel
dd x L(?, ¦×, ¦×)

Let us consider a general action S [?, ¦×, ¦×] = (2.1)

for nB real scalar ?elds ?i and nF spin 1/2 ?elds ¦×a . Anticipating the later use of dimensional regularization, I am starting in d-dimensional Minkowski space. To construct the

1

generating functional Z of Green functions, these ?elds are coupled to external sources ji (i = 1, . . . , nB ), ¦Ña , ¦Ña (a = 1, . . . , nF ), Z [j, ¦Ñ, ¦Ñ] = eiW [j,¦Ñ,¦Ñ] = [d?d¦×d¦×] ei(S [?,¦×,¦×]+j
T ?+¦×¦Ñ+¦Ñ¦× )

,

(2.2)

where W [j, ¦Ñ, ¦Ñ] is the generating functional of connected Green functions. I have used the notation (2.3) j T ? + ¦×¦Ñ + ¦Ñ¦× := dd x (ji ?i + ¦× a ¦Ña + ¦Ña ¦×a ) . The normalization of the functional integral is determined by the condition Z [0, 0, 0] = 1. The solutions of the classical equations of motion ¦ÄS + ji = 0, ¦Ä?i ¦ÄS + ¦Ña = 0, ¦Ä¦× a ¦ÄS ? ¦Ña = 0 ¦Ä¦×a (2.4)

are denoted by ?cl , ¦×cl . They are uniquely determined functionals of the external sources. With ?uctuation ?elds ¦Î, ¦Ç de?ned by ?i = ?cl,i + ¦Îi , ¦×a = ¦×cl,a + ¦Ça , (2.5)

the integrand in (2.2) is expanded in terms of ¦Î, ¦Ç, ¦Ç. The resulting loop expansion of the generating functional W = WL=0 + WL=1 + . . . starts with the classical action in the presence of external sources: WL=0 = S [?cl , ¦×cl , ¦× cl ] + j T ?cl + ¦× cl ¦Ñ + ¦Ñ¦×cl . The one-loop term WL=1 is given by a Gaussian functional integral eiWL=1 = where S (2) [?cl , ¦×cl , ¦× cl ; ¦Î, ¦Ç, ¦Ç] = dd x L(2) (?cl , ¦×cl , ¦× cl ; ¦Î, ¦Ç, ¦Ç) (2.8) is quadratic in the ?uctuation variables. Employing the notation introduced in (2.3), S (2) takes the general form S (2) = 1 T ¦Î A¦Î + ¦ÇB¦Ç + ¦Î T ¦£¦Ç + ¦Ç ¦£¦Î 2 1 T T = ¦Î A¦Î + ¦ÇB¦Ç ? ¦Ç T B T ¦Ç T + ¦Î T ¦£¦Ç ? ¦Ç T ¦£ ¦Î + ¦Ç ¦£¦Î ? ¦Î T ¦£T ¦ÇT 2 [d¦Îd¦Çd¦Ç] eiS
(2) [? cl ,¦×cl ,¦× cl ;¦Î,¦Ç,¦Ç ]

(2.6)

,

(2.7)

,

(2.9)

where A, B, ¦£, ¦£ are operators in the respective spaces; A = AT and B are bosonic di?erential operators, whereas ¦£ and ¦£ are fermionic (Grassmann) operators. They all depend on the classical solutions ?cl , ¦×cl . 2

The standard procedure for the evaluation of (2.7) is to integrate ?rst over the fermion ?elds ¦Ç, ¦Ç to yield the bosonic functional integral eiWL=1 = det B This leads to the familiar result WL=1 = i T ln det(A ? ¦£B ?1 ¦£ + ¦£T B ?1T ¦£ ) ? ln det A0 ? i(ln det B ? ln det B0 ) 2 i B i A T = ? i Tr ln + Tr ln(1 ? A?1 ¦£B ?1 ¦£ + A?1 ¦£T B ?1T ¦£ ) Tr ln 2 A0 B0 2 ¡Þ i i A B T n = Tr ln ? i Tr ln ? Tr A?1 ¦£B ?1 ¦£ ? A?1 ¦£T B ?1T ¦£ , 2 A0 B0 n=1 2n A0 := A|j =¦Ñ=¦Ñ=0 , B0 := B |j =¦Ñ=¦Ñ=0 . (2.10) [d¦Î ] e 2 ¦Î
T i T (A?¦£B ?1 ¦£+¦£T B ?1T ¦£ )¦Î

.

Recalling that A?1 , B ?1 are the scalar and fermion matrix propagators in the presence of external sources, the one-loop functional WL=1 is seen to be a sum of the bosonic one-loop i functional 2 Tr ln(A/A0 ), the fermion-loop functional ?i Tr ln(B/B0 ) and a mixed oneloop functional where scalar and fermion propagators alternate. In order to determine the ultraviolet divergences that occur in the mixed term in (2.10), the calculational inconveniences mentioned in Sect. 1 are encountered. These problems can be circumvented [7] by reorganizing the three parts of WL=1 into a more compact form, using the notion of supermatrices, supertraces, etc. (see for instance [8, 9]). Combining the bosonic and fermionic ?uctuation variables in a multicomponent ?eld ? ? ¦Î ? ? ¦Ë=? ¦Ç ? , (2.11) ¦ÇT S (2) in (2.9) can be written as 1 S (2) = ¦ËT K ¦Ë . 2 (2.12)

The one-loop functional of connected Green functions can now be written in compact form [7] in terms of a supertrace WL=1 = i K . Str ln 2 K0 3

The explicit form of the supermatrix operator K follows immediately from the second line in (2.9): ? ? ¦£ ?¦£T A ? T (2.13) K=? ? ?¦£ 0 ?B T ? . ¦£ B 0

(2.14)

With the notation Str O = dd x str x|O |x

I distinguish supertraces with and without space-time integration. For actual calculations, the form of the supermatrix operator K de?ned in (2.13) is not the most convenient one. Applying a similarity transformation to K , the generating functional can also be written as WL=1 = with
?

i K¡ä Str ln ¡ä 2 K0 ¡Ì
? ¡Ì ? ¦£ ? ? ¦£T ? ?B 0 ? . T 0 ?B

(2.15)

The arbitrary mass parameter ? introduced in (2.16) guarantees equal dimensions for all entries in K ¡ä ([K ¡ä ] = [A] = 2). Although this quantity does not, of course, appear in any ?nal result, it turns out to be quite helpful for the inspection of expressions at intermediate stages of calculations. In the proper-time formulation, the one-loop functional assumes the form WL=1 = ? i 2 i = ? 2
¡Þ 0 ¡Þ 0

A ? ¡Ì? ¦£ ¡ä K =? ¡Ì T ?¦£

(2.16)

d¦Ó ¡ä ¡ä Str ei¦Ó K ? ei¦Ó K0 ¦Ó d¦Ó ¡ä ¡ä dd x str x|ei¦Ó K ? ei¦Ó K0 |x , ¦Ó

(2.17)

which is just the desired super-heat-kernel representation. Note that the convergence of the integral at the upper end (¦Ó ¡ú ¡Þ) is guaranteed by the small imaginary parts present in the bosonic and fermionic di?erential operators A and B , which are ensuring at the same time the usual Feynman boundary conditions. (For a free theory A = ?2 ? M 2 + i?, B = i ? ? m + i?.) On the other hand, the behaviour of the integral at the lower end exhibits the divergence structure of the theory under investigation. As long as we are only interested in those parts of the one-loop functional that are at most bilinear in fermion ?elds, the supermatrix K ¡ä can be reduced to the simpler form ¡Ì 2? ¦£ A ¡ä¡ä K = ¡Ì , (2.18) 2? ¦£ ?B such that the one-loop functional reads WL=1 = K ¡ä¡ä i B i + ... Str ln ¡ä¡ä ? Tr ln 2 K0 2 B0 (2.19)

The terms omitted are at least quartic in the fermion ?elds. 4

3

Scalars Interacting with Heavy Fermions

In the case of chiral perturbation theory with heavy baryons, the ?uctuation action (2.9) generated by the lowest order meson¨Cbaryon Lagrangian (O(p2 ) in the mesonic and O(p) in the baryonic part) has the general form 1 S (2) = ? ¦Î T (D? D ? + Y )¦Î + ¦Ç(¦Á + ¦Â? D ? )¦Î + ¦Î T (¦Ä ? ¦Â ? D ? )¦Ç + ¦Çiv? D ? ¦Ç , 2 where D? = ?? + X? , D? = ?? + f? , (3.1)

X? , Y and f? are bosonic (matrix-) ?elds, whereas ¦Á and ¦Â? are fermionic objects. The form of ¦Ä in (3.2) is required by the reality of (3.1). v? is the usual velocity vector introduced in the heavy mass expansion. Apart from the condition v ¡¤ ¦Â = 0 (which is indeed satis?ed in heavy baryon chiral perturbation theory), no further assumption about the terms entering in (3.1) is made in this section. The discussion will therefore apply to a rather large class of theories of scalars interacting with heavy fermions, not necessarily related to chiral perturbation theory. The further specialization to the pion¨C nucleon system is reserved until the next chapter. The action (3.1) is invariant under local gauge transformations ¦Î (x) ¦Ç (x) X? Y f? ¦Á ¦Â? ¡ú ¡ú ¡ú ¡ú ¡ú ¡ú ¡ú R(x)¦Î (x) , R(x)T R(x) = 1 , U (x)¦Ç (x) , U (x)? U (x) = 1 , R?? R?1 + RX? R?1 , RY R?1 , U?? U ?1 + Uf? U ?1 , U¦ÁR?1 , U¦Â? R?1 .

¦Ä = ¦Á ? D? ¦Â ? , v2 = 1 , v¡¤¦Â =0 .

(3.2)

(3.3)

Consequently, also the divergent part of the one-loop functional exhibits this symmetry property [10]. The matrix-?elds Y , ¦Á, ¦Â? together with their covariant derivatives ?? Y := ?? Y + [X? , Y ], (3.4)

?? ¦Â¦Í := ?? ¦Â¦Í + f? ¦Â¦Í ? ¦Â¦Í X? , and the associated ¡°?eld-strength¡± tensors X?¦Í := ?? X¦Í ? ?¦Í X? + [X? , X¦Í ], f?¦Í := ?? f¦Í ? ?¦Í f? + [f? , f¦Í ]

?? ¦Á := ?? ¦Á + f? ¦Á ? ¦ÁX? ,

(3.5)

are therefore the appropriate building blocks for the construction of a gauge-invariant action. 5

The general heat-kernel formalism of the preceding section will now be applied to (3.1). In this case, the matrix-operators A, B , ¦£ and ¦£ de?ned in (2.9) are given by A = ?D 2 ? Y , B = iv ¡¤ D , ¦£=¦Á+¦Â¡¤D , ¦£= ¦Ä?¦Â¡¤D . (3.6)

As I am considering only terms at most bilinear in the fermionic variables, the form (2.18) for the supermatrix operator is the appropriate one. Employing the method of Ball [11], the relevant diagonal space-time matrix element can be written as str x|ei¦Ó K |x
¡ä¡ä

= str = str

dd k x|ei¦Ó K |k k |x = str dd k i¦Ó K ¡ä¡ä e 1, (2¦Ð )d ¡Ì

¡ä¡ä

dd k ikx i¦Ó K ¡ä¡ä ?ikx e e e (2¦Ð )d (3.7)

with K =
¡ä¡ä

The further evaluation of this expression is considerably simpli?ed by the observation that in the following intermediate steps we may restrict ourselves to constant ?elds [11] X? , ¦Á, ¦Â? , f? , Y = ?X 2 . As the ?nal result for the one-loop divergences has to be gaugeinvariant, no information is lost and the full expression for space-time dependent ?elds is recovered by the substitutions ? X2 ¡ú Y , ?[X? , X 2] ¡ú ?? Y , [X? , X¦Í ] ¡ú X?¦Í ,

2 2 ? ¡ÌD ? Y + k + 2ik ¡¤ D 2?(¦Á + ¦Â ¡¤ D ? ik ¡¤ ¦Â )

2?(¦Ä ? ¦Â ¡¤ D + ik ¡¤ ¦Â ) ?(iv ¡¤ D + v ¡¤ k )

.

(3.8)

f? ¦Â¦Í ? ¦Â¦Í X? ¡ú ?? ¦Â¦Í , [f? , f¦Í ] ¡ú f?¦Í . In this approach, (3.7) reduces to the much simpler expression str x|ei¦Ó K |x = with M = i¦Ó k 2 + 2ik ¡¤ X 0 0 ?(iv ¡¤ f + v ¡¤ k ) , .
¡ä¡ä

f? ¦Á ? ¦ÁX? ¡ú ?? ¦Á ,

(3.9)

dd k str eM +N (2¦Ð )d

(3.10)

N = i¦Ó 2?

0 ¦Ä ? ¦Â ¡¤ f + ik ¡¤ ¦Â ¦Á + ¦Â ¡¤ X ? ik ¡¤ ¦Â 0

(3.11)

Let us ?rst consider the part bilinear in the fermionic matrix N (generating the terms of the form ¦Á . . . ¦Á, ¦Á . . . ¦Â? , ¦Â ? . . . ¦Á, ¦Â ? . . . ¦Â¦Í ). The corresponding part of the generating functional (2.10) is just WL=1 |¦£...¦£ := ?i Tr (A?1 ¦£B ?1 ¦£) . 6 (3.12)

The appropriate decomposition of the exponential in (3.10) can be performed by using Feynman¡¯s ¡°disentangling¡± theorem [12]: exp(M + N ) = exp M Ps exp with N (s) := e?sM NesM and Ps exp
1 0 ¡Þ 1 s1 0 sn?1 0 1 0

ds N (s)

(3.13)

ds N (s) :=

n=0 0

ds1

ds2 . . .

dsn N (s1 )N (s2 ) . . . N (sn ) .

(3.14)

(In the mathematical literature, (3.13) is also known as ¡°Duhamel¡¯s formula¡±.) Picking out the part bilinear in N , str eM +N =
1 0 s

ds

0

ds¡ä str e(1?s)M Ne(s?s )M Nes M + . . . ,

¡ä

¡ä

(3.15)

a few simple manipulations lead to str eM +N = ?2?¦Ó 2
1 0

dz ei¦Ó [(1?z )k

2 +z?v ¡¤k ]

tr (¦Ä ? ¦Â ¡¤ f + i¦Â ¡¤ k )e?¦Ó z?v¡¤f (3.16)

(¦Á + ¦Â ¡¤ X ? i¦Â ¡¤ k )e?2¦Ó (1?z )k¡¤X + . . .

After integration over z , the ?-dependent terms cancel once the proper-time and the momentum-space integrals are applied. The remaining contribution to WL=1 assumes the form WL=1 |¦£...¦£ = ? dd x
¡Þ 0

dt 3?d t t

dd l iv¡¤l e tr (¦Ä ? ¦Â ¡¤ f + i¦Â ¡¤ l/t)e?tv¡¤f (2¦Ð )d (3.17)

(¦Á + ¦Â ¡¤ X ? i¦Â ¡¤ l/t)(l2 + 2itl ¡¤ X )?1 ,

where a suitable change of the integration variables has been performed. The divergent part (for d ¡ú 4) can now be easily isolated:
div WL =1 |¦£...¦£ = ¦£(4 ? d)

dd x

dd l eiv¡¤l (2¦Ð )d l2

tr

1 ¦Â ¡¤ l (v ¡¤ f )3 ¦Â ¡¤ l 3! + (¦Ä ? ¦Â ¡¤ f )(¦Á + ¦Â ¡¤ X ) + i(¦Ä ? ¦Â ¡¤ f ) v ¡¤ f ¦Â ¡¤ l (¦Ä ? ¦Â ¡¤ f ) v ¡¤ f (¦Á + ¦Â ¡¤ X ) + ?i¦Â ¡¤ l v ¡¤ f (¦Á + ¦Â ¡¤ X ) +

1 2il ¡¤ X ¦Â ¡¤ l (v ¡¤ f )2 ¦Â ¡¤ l 2! l2 4(l ¡¤ X )2 (l 2 )2 (3.18)

+ ?i(¦Ä ? ¦Â ¡¤ f ) ¦Â ¡¤ l + i¦Â ¡¤ l (¦Á + ¦Â ¡¤ X ) ? ¦Â ¡¤ l v ¡¤ f ¦Â ¡¤ l ?¦Â ¡¤ l ¦Â ¡¤ l 8i(l ¡¤ X )3 (l 2 )3 . 7

The necessary formulas for the l-integration are given in the Appendix. In the last step, one has to identify the appropriate gauge-invariant combinations (constituting a nontrivial check of the calculation) and reconstruct the full result by using (3.9). In this way, I ?nally obtain:
div WL =1 |¦£...¦£ =

i 48¦Ð 2 (d ? 4)

d4 x tr

?12 ¦Á v ¡¤ ?¦Á + 6 ¦Á¦Â? X ?¦Í v¦Í + ¦Â ? ¦ÁX ?¦Í v¦Í . (3.19)

+6 ¦Â ? (v ¡¤ ?¦Â¦Í )X ?¦Í + 4 ¦Â ? ¦Â¦Í v ¡¤ ?X ?¦Í + 2 ¦Â ? ¦Â¦Í ?? X ¦Í¦Ñ v¦Ñ

?3 ¦Â ¡¤ ¦Â v ¡¤ ?Y + 2 ¦Â ? (v ¡¤ ?¦Â ? )Y ? 4¦Â ? (v ¡¤ ?)3 ¦Â ? + ¦Â ¡¤ ¦Â ?? X ?¦Í v¦Í

Note that (3.19) has to be real, which is another independent check of the result. The remaining part of the generating functional with the fermionic operators ¦£, ¦£ turned o?, i B A WL=1 |¦£=¦£=0 = Tr ln ? i Tr ln , (3.20) 2 A0 B0 does not require any additional e?ort. A simple calculation (involving a Gaussian momentum-space integration) gives i 1 Tr ln A|div = ? 2 2 (4¦Ð ) (d ? 4) d4 x tr 1 1 X?¦Í X ?¦Í + Y 2 12 2 , (3.21)

which is the standard result obtained by ¡¯t Hooft [10] using diagrammatic methods. The second term in (3.20) vanishes identically, as it corresponds to the closed loop of a ¡°light¡± fermion component in the heavy mass expansion: Tr ln B = ? = ? which follows from dd l iv¡¤l e = ¦Ä (d) (v ) = 0 . d (2¦Ð ) dd x dd x
¡Þ 0 ¡Þ 0

dt dd k tr eit(iv¡¤D+v¡¤k) 1 d t (2¦Ð ) dd l iv¡¤l dt ?d t e tr e?tv¡¤D 1 = 0 , t (2¦Ð )d

4

Renormalization of the Pion¨CNucleon Interaction

The functionals (3.19) and(3.21) are the basic formulas for the analysis of the one-loop divergences to O(p3 ) in heavy baryon chiral perturbation theory. They can be applied to both the two-?avour and the three-?avour case. In the following I shall con?ne myself to chiral SU (2). The starting point for the formulation of the e?ective ?eld theory of the pion¨Cnucleon system is QCD with the two light ?avours u, d coupled to external Hermitian ?elds [13]: 1 s + ¦Ã5 A? q ? q ?(S ? i¦Ã5 P )q , L = L0 ?¦Ã ? V? + V? QCD + q 3 8 q= u d . (4.1)

L0 QCD is the QCD Lagrangian with mu = md = 0, S and P are general two-dimensional matrix ?elds, the isotriplet vector and axial-vector ?elds V? , A? are traceless and the s isosinglet vector ?eld V? is included to generate the electromagnetic current. Explicit chiral symmetry breaking is built in by setting S = Mquark = diag [mu , md ]. The chiral group G = SU (2)L ¡Á SU (2)R is spontaneously broken to the isospin group SU (2)V . It is realized non-linearly [14] on the Goldstone pion ?elds ¦Õ: uR (¦Õ) ¡ú gR uR (¦Õ)h(g, ¦Õ) , uL (¦Õ) ¡ú gL uL (¦Õ)h(g, ¦Õ)?1,
g ?1 g

g = (gL , gR ) ¡Ê G ,

(4.2)

where uL , uR are elements of the chiral coset space SU (2)L ¡Á SU (2)R /SU (2)V and the compensator ?eld h(g, ¦Õ) is in SU (2)V . The nucleon doublet ¦· transforms as ¦·= p n ¡ú ¦·¡ä = h(g, ¦Õ)¦·
g

(4.3)

under chiral transformations. The local nature of this transformation requires a connection 1 ? ¦£? = uR (?? ? ir? )uR + u? (4.4) L (?? ? i?? )uL 2 in the presence of external gauge ?elds r? = V? + A? , to de?ne a covariant derivative
s ?? ¦· = (?? + ¦£? ? iV? )¦· .

?? = V? ? A?

(4.5)

(4.6)

To lowest order in the chiral expansion the e?ective Lagrangian of the pion¨Cnucleon system is [1, 13] Le? = with
? u? = i u? R (?? ? ir? )uR ? uL (?? ? i?? )uL , ? ? ¦Ö+ = u? R ¦ÖuL + uL ¦Ö uR .

F2 ? i ? ? m + gA u¦Ã5 )¦· , u? u? + ¦Ö+ + ¦·( 4 2

(4.7)

¦Ö = 2B (S + iP ),

F, m, gA are the pion decay constant, the nucleon mass and the neutron decay constant in the chiral limit, whereas B is related to the quark condensate. . . . stands for the trace in ?avour space. The heavy baryon mass expansion of (4.7) is obtained by introducing velocitydependent ?elds
+ Nv (x) = eimv¡¤x Pv ¦·(x) , imv¡¤x ? Hv (x) = e Pv ¦·(x) , 1 ¡À (1¡À v ) , v2 = 1 , Pv = 2

(4.8)

9

leading to Le? = F2 u?u? + ¦Ö+ + N v (iv ¡¤ ? + gA S ¡¤ u)Nv + . . . 4 (4.9)

The additional terms involving the ¡°heavy¡± fermion components Hv are irrelevant for our present purposes. (For a more detailed discussion the reader is referred to [5, 15].) The only dependence on Dirac matrices in (4.9) is through the spin-vector matrices i S ? = ¦Ã5 ¦Ò ?¦Í v¦Í , 2 S¡¤v =0 , 3 S2 = ? 1 , 4 (4.10)

which obey the (anti-) commutation relations 1 {S ? , S ¦Í } = (v ? v ¦Í ? g ?¦Í ) , 2 [S ? , S ¦Í ] = i¦Å?¦Í¦Ñ¦Ò v¦Ñ S¦Ò .
(2)

(4.11)

To obtain the associated second-order ?uctuation Lagrangian Le? , (4.9) is expanded around the classical ?elds ¦Õcl , Nv,cl . In the standard ¡°gauge¡± uR (¦Õcl ) = u? L (¦Õcl ) =: u(¦Õcl ) a convenient choice of the bosonic ?uctuation variables ¦Îi (i = 1, 2, 3) is given by [13] uR (¦Õ) = u(¦Õcl ) e
¡ú
¡ú ¡ú i ¦Î (¦Õ)¡¤ ¦Ó 2F

,

uL (¦Õ) = u (¦Õcl ) e

?

?

¡ú ¡ú i ¦Î (¦Õ)¡¤ ¦Ó 2F

,

¡ú

¦Î (¦Õcl ) = 0 ,

(4.12)

where ¦Ó denotes the Pauli matrices. For the fermion ?elds I write Nv = Nv,cl + ¦Ç . In this way I get Le?
(2)

=

1 (d? ki ¦Îi )(d? kj ¦Îj ) ? ¦Òij ¦Îi ¦Îj 2 1 + 2 N v [i¦Îi [¦Ói , ¦Ók ](v ¡¤ dkj ¦Îj ) + gA ¦Îi [¦Ói , [S ¡¤ u, ¦Ój ]]¦Îj ]Nv 8F i 1 + N v [v ¡¤ u, ¦Ói]¦Îi ? gA S? ¦Ói (d? ij ¦Îj ) ¦Ç F 4 i 1 + ¦Ç [v ¡¤ u, ¦Ói ]¦Îi ? gA S? ¦Ói (d? ij ¦Îj ) Nv F 4 +¦Ç(iv ¡¤ ? + gA S ¡¤ u)¦Ç , 1 ? ¦£ [¦Ói , ¦Ój ] , 2

(4.13)

where
? ? d? ij = ¦Äij ? + ¦Ãij , ? ¦Ãij =?

¦Òij =

1 (u ¡¤ u + ¦Ö+ )¦Äij ? ¦Ói u? ¦Ój u? . 4

(4.14)

Note that the quantities Nv , u? , ¦£? , ¦Ö+ in (4.13) are to be taken at the solutions of the classical equations of motion. (The subscript ¡°cl¡± has only been dropped for simplicity.)

10

It is now easy to verify that the action associated with (4.13) can indeed be written in the standard form (3.1) by setting X? = ¦Ã ? + g ? , Y = iv ? ? i = 1, 2 , 3 , gij = ? 2 N v [¦Ói , ¦Ój ]Nv , 8F gA ¦Ò+s , sij = N v (2 ¦Äij S ¡¤ u ? ¦Ói S ¡¤ u ¦Ój ? ¦Ój S ¡¤ u ¦Ói) Nv , 4F 2 s ¦£? ? iV? ? iv? gA S ¡¤ u i ([v ¡¤ u, ¦Ói ]Nv )a , a = 1, 2 , 4F gA ? S? (¦Ói Nv )a . (4.15) F

f? = ¦Áai = (¦Â? )ai =

Let us ?rst consider the one-loop divergences generated by (3.21). Using (4.15), tr Y 2 = tr ¦Ò 2 + 2 tr (¦Òs) + . . . and tr (X?¦Í X ?¦Í ) = tr ¦Ã?¦Í ¦Ã ?¦Í + 4 tr {¦Ã?¦Í (? ? g ¦Í + [¦Ã ? , g ¦Í ])} + . . . , where ¦Ã?¦Í := ?? ¦Ã¦Í ? ?¦Í ¦Ã? + [¦Ã? , ¦Ã¦Í ] . (4.18) The ?rst terms on the right-hand sides of (4.16) and (4.17) are purely mesonic; they determine the divergence structure of the well-known Gasser¨CLeutwyler functional of O(p4 ) [13]. The second ones are bilinear in the fermion ?elds, whereas the dots refer to irrelevant terms ? (N v . . . Nv )2 . To facilitate the comparison with [5], I write the fermion bilinears extracted from (3.21) in the following form:
div WL =1 |¦£=¦£=0 =

(4.16)

(4.17)

d4 x N v ¦²div 1 Nv ,

¦²div = ? 1

1 8 ¦Ð 2 F 2 (d ? 4)

¦²1 .

(4.19)

For ¦²1 I ?nd gA i ( u ¡¤ u + ¦Ö+ S ¡¤ u + S ¡¤ u u? u?) , ¦²1 = ? (?? ¦£?¦Í v ¦Í ) + 6 4 where ¦£?¦Í := ?? ¦£¦Í ? ?¦Í ¦£? + [¦£? , ¦£¦Í ] . (4.21) This result agrees with the corresponding expression in (36) of [5]. Note that I have used several SU (2) relations to arrive at a simpler form for ¦²1 in (4.20). The one-loop divergences originating from (3.19) are again presented in the form
div WL =1 |¦£...¦£ =

(4.20)

d4 x N v ¦²div 2 Nv ,

¦²div 2 = ? 11

1 8 ¦Ð 2 F 2 (d ? 4)

¦²2 .

(4.22)

Inserting (4.15) in (3.19), I obtain: 1 1 1 [2(v ¡¤ u)2 + (v ¡¤ u)2 ]v ¡¤ ? + v ¡¤ u(v ¡¤ ?v ¡¤ u) + v ¡¤ u(v ¡¤ ?v ¡¤ u) 4 2 4 1 1 + gA ? v ¡¤ u S ¡¤ u v ¡¤ u + S ¡¤ u(v ¡¤ u)2 ? S ? v ¦Í [¦£?¦Í , v ¡¤ u] 2 4 5 3 2 + igA ? (v ¡¤ ?)3 ? (?? ¦£?¦Í v ¦Í ) + i¦Å?¦Í¦Ñ¦Ò v¦Ñ S¦Ò [2¦£?¦Í v ¡¤ ? + (v ¡¤ ?¦£?¦Í )] 2 6 3 3 4u ¡¤ u + 3¦Ö+ v ¡¤ ? ? (v ¡¤ ? 4u ¡¤ u + 3¦Ö+ ) ? 32 16 1 1 3 + gA ? S ¡¤ u(v ¡¤ ?)2 ? 2S ? S ¡¤ u ¦£?¦Í S ¦Í ? (v ¡¤ ?S ¡¤ u)v ¡¤ ? 2 2 1 1 1 ? (v ¡¤ ?)2 S ¡¤ u ? u? u? S ¡¤ u ? S ¡¤ u ¦Ö+ 6 4 16 2 4 + igA S? [2(S ¡¤ u)2 ? 4 (S ¡¤ u)2 ]v ¡¤ ? + (v ¡¤ ?S ¡¤ u)S ¡¤ u 3 4 + S ¡¤ u(v ¡¤ ?S ¡¤ u) ? 4 S ¡¤ u(v ¡¤ ?S ¡¤ u) S ? 3 4 2 5 + gA S? (S ¡¤ u )3 ? (S ¡¤ u )3 S ? , (4.23) 3 3 which is in agreement with the corresponding result in [5]. (Note that ¡°+?¡± in the fourth line of (53) in [5] should be read as a minus sign.) ¦²2 = i

5

Conclusions

I have shown that the super-heat-kernel technique constitutes the appropriate theoretical tool for analyzing the one-loop divergences in systems with (non-renormalizable) boson¨C fermion interactions. I recall here the essential ingredients that were combined to arrive at an e?cient computational scheme: ? The one-loop functional is written in terms of the superdeterminant of a suitably chosen supermatrix operator. ? The associated super-heat-kernel representation is the appropriate form of the oneloop functional for studying its divergence structure. ? It is easier to determine the diagonal heat-kernel matrix elements directly by inserting a complete set of plane waves instead of calculating the Seeley¨CDeWitt coe?cients with two di?erent space-time arguments and taking the coincidence-limit at the end. ? The heat-kernel-representation is perfectly well de?ned also for supermatrices with ?rst-order (fermion) di?erential operators1 .
Note, however, that ¡°squaring¡± of the fermionic di?erential operator may simplify the analysis in theories where the full relativistic Dirac operator is still present [7].
1

12

? The second-order ?uctuation action is invariant under a local gauge transformation. As a consequence, this symmetry property is also shared by the divergence functional. ? At intermediate stages, the calculation can be carried out with constant (classical) ?elds, avoiding cumbersome manipulations with derivatives acting on space-time dependent objects. At the end, the general result is recovered by gauge invariance. ? Feynman¡¯s disentangling theorem allows the proper decomposition of the exponential of a sum of non-commuting terms. ? With the divergence functional given in compact form, the one-loop renormalization of e?ective quantum ?eld theories becomes an easy task, requiring only a few purely algebraic operations. The application to heavy baryon chiral perturbation theory with two ?avours served as an explicit example. A previous result for the counterterms to O(p3 ) was con?rmed. With the super-heat-kernel method at hand, the systematic study of e?ective ?eld theories at the one-loop level is simpli?ed considerably. I am giving here a small selection of possible applications and extensions of the present work: ? The treatment of the meson¨Cbaryon interaction with three ?avours is completely analogous to the two-?avour case disussed before. ? The inclusion of ?elds with higher spin (photon, ?-resonance, etc.) is straightforward. Their components are simply added to the bosonic and fermionic sectors, respectively. ? The completion of the one-loop renormalization for the pion¨Cnucleon interaction up to O(p4 ) may be achieved by a suitable extension of (3.1). ? For the analysis of fermionic bound states, the complete form (2.16) of the supermatrix operator must be used, as terms quartic in the fermion ?elds are relevant in this case. ? In analogy to the mesonic case [13], the super-heat-kernel representation might also be useful for the ?nite part of the one-loop functional with two external baryons.

Acknowledgements
I am indebted to Gerhard Ecker, J¡§ urg Gasser, Joachim Kambor and Marc Knecht for helpful discussions and useful comments.

13

Appendix
I consider ?rst the integrals In (v 2 ) :=
2n?d dd l eiv¡¤l = fn (d)(v 2) 2 d 2 n (2¦Ð ) (l + i?)

(A.1)

with an arbitrary four-vector v? . The fn (d) are given by f1 (d) = (?i)d?1 and fn (d) = 2 n ? 1 (n ¦£(d ? 2)
d? 1 2

(4¦Ð )

1 ¦£( d? ) 2

(A.2)

The momentum space integrals occurring in (3.18) are now obtained by di?erentiating (A.1) a su?cient number of times with respect to v? and setting v 2 = 1 at the end: dd l eiv¡¤l (2¦Ð )d l2 dd l eiv¡¤l l? l¦Í (2¦Ð )d l2 dd l eiv¡¤l l? (2¦Ð )d (l2 )2 dd l eiv¡¤l l? l¦Í (2¦Ð )d (l2 )2 dd l eiv¡¤l l? l¦Í l¦Ñ (2¦Ð )d (l2 )2 dd l eiv¡¤l l? l¦Í l¦Ñ (2¦Ð )d (l2 )3 dd l eiv¡¤l l? l¦Í l¦Ñ l¦Ò (2¦Ð )d (l2 )3 ?¡ú ?¡ú ?¡ú ?¡ú ?¡ú ?¡ú ?¡ú
d¡ú4 d¡ú4 d¡ú4 d¡ú4 d¡ú4 d¡ú4 d¡ú4

1 f1 (d) , ? 1)! (d ? 2n) . . . (d ? 4)

n = 2, 3 , . . .

(A.3)

dd l eiv¡¤l d¡ú4 l? l¦Í l¦Ñ l¦Ò l¦Ó ?¡ú d 2 4 (2¦Ð ) (l )

i , (A.4) 4¦Ð 2 i (g?¦Í ? 4v? v¦Í ) , (A.5) 2¦Ð 2 v? ? 2 , (A.6) 8¦Ð i (g?¦Í ? 2v? v¦Í ) , (A.7) 8¦Ð 2 1 [? (g?¦Í v¦Ñ + . . .) + 4v? v¦Í v¦Ñ ] , (A.8) 4¦Ð 2 1 [? (g?¦Í v¦Ñ + . . .) + 2v? v¦Í v¦Ñ ] , (A.9) 32¦Ð 2 i [(g?¦Í g¦Ñ¦Ò + . . .) ? 2 (g?¦Í v¦Ñ v¦Ò + . . .) 32¦Ð 2 +8v? v¦Í v¦Ñ v¦Ò ] , (A.10) 1 [? (g?¦Í g¦Ñ¦Ò v¦Ó + . . .) + 2 (g?¦Í v¦Ñ v¦Ò v¦Ó + . . .) 192¦Ð 2 ?8v? v¦Í v¦Ñ v¦Ò v¦Ó ] . (A.11)

The dots indicate the necessary symmetrizations.

14

References
¡¦ [1] J. Gasser, M.E. Sainio and A. Svarc, Nucl. Phys. B 307 (1988) 779. [2] A. Krause, Helv. Phys. Acta 63 (1990) 3. [3] E. Jenkins and A.V. Manohar, Phys. Lett. B 255 (1991) 558. [4] N. Isgur and M. Wise, Phys. Lett. B 232 (1989) 113; ibid. B 237 (1990) 527; B. Grinstein, Nucl. Phys. B 339 (1990) 253; E. Eichten and B. Hill, Phys. Lett. B 234 (1990) 511; H. Georgi, Phys. Lett. B 240 (1990) 447. [5] G. Ecker, Phys. Lett. B 336 (1994) 508. [6] G. M¡§ uller and U.-G. Mei?ner, Nucl. Phys. B 492 (1997) 379. [7] H. Neufeld, J. Gasser and G. Ecker, The One-Loop Functional as a Berezinian, CERN-TH/98-197 (hep-ph/9806436). [8] R. Arnowitt, P. Nath and B. Zumino, Phys. Lett. B 56 (1975) 81. [9] F.A. Berezin, ¡°Introduction to Superanalysis¡±, A.E. Kirillov, Ed. (D. Reidel Publishing Company, Dordrecht, 1987). [10] G. ¡¯t Hooft, Nucl. Phys. B 62 (1973) 444. [11] R.D. Ball, Phys. Rep. 182 (1989) 1. [12] R.P. Feynman, Phys. Rev. 84 (1951) 108. [13] J. Gasser and H. Leutwyler, Ann. Phys. 158 (1984) 142. [14] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247. [15] G. Ecker, M. Moj¡¦ zi¡¦ s, Phys. Lett. B 365 (1996) 312.

15

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