Dimensional regularization for N=1supersymmetric sigma models and the worldline formalism

Dimensional regularization for

NÄ1 supersymmetric sigma models and the worldline formalism

Fiorenzo Bastianelli,*Olindo Corradini,†and Andrea Zirotti‡

Dipartimento di Fisica, Universita` di Bologna and INFN, Sezione di Bologna via Irnerio 46, I-40126 Bologna, Italy 共Received 28 February 2003; published 19 May 2003兲

We generalize the worldline formalism to include spin 1/2 fields coupled to gravity. To this purpose we first extend dimensional regularization to supersymmetric nonlinear sigma models in one dimension. We consider a finite propagation time and find that dimensional regularization is a manifestly supersymmetric regularization scheme, since the classically supersymmetric action does not need any counterterm to preserve worldline supersymmetry. We apply this regularization scheme to the worldline description of Dirac fermions coupled to gravity. We first compute the trace anomaly of a Dirac fermion in 4 dimensions, providing an additional check on the regularization with finite propagation time. Then we come to the main topic and consider the one-loop effective action for a Dirac field in a gravitational background. We describe how to represent this effective action as a worldline path integral and compute explicitly the one- and two-point correlation functions, i.e. the spin 1/2 particle contribution to the graviton tadpole and graviton self-energy. These results are presented for the general case of a massive fermion. It is interesting to note that in the worldline formalism the coupling to gravity can be described entirely in terms of the metric, avoiding the introduction of a vielbein. Consequently, the fermion-graviton vertices are always linear in the graviton, just like the standard coupling of fermions to gauge fields.

DOI: 10.1103/PhysRevD.67.104009 PACS number共s兲: 04.62.⫹v

I. INTRODUCTION

One dimensional supersymmetric nonlinear sigma models are useful to describe in first quantization the propagation of Fermionic particles in a curved background. In fact, it is well known thatN⫽1 supersymmetric sigma models describe the worldline dynamics of a spinning particle关1兴. Mastering the path integral quantization of such models provides a useful tool for treating spin 1/2 particles coupled to gravity. The purpose of this paper is twofold. We first extend dimensional regularization to supersymmetric sigma models with finite propagation 共proper兲 time. Then, with this regularization scheme at hand, we generalize the worldline formalism to include spin 1/2 fields coupled to gravity. This extends the scalar particle case treated in Ref. 关2兴. The resulting Feyn-man rules are simpler than the standard ones obtained from the second quantized action. In particular, the fermion-graviton vertices can always be taken linear in the fermion-graviton field, a fact which seems to point once more to unexpected perturbative relations between gravity and gauge theories, as reviewed in Ref.关3兴.

Path integrals for supersymmetric sigma models in one dimension were originally used for deriving formulas for in-dex theorems and chiral anomalies关4–6兴. However, for ob-taining those results the details of how to properly define and regulate the path integrals at higher loops were not necessary. Due to the worldline supersymmetry the chiral anomalies are seen as a topological quantity, the Witten index关7兴, which is independent of␤, the propagation time in the sigma model. Thus a semiclassical approximation共which consists in calcu-lating a few determinants兲 already gives the complete results.

The quantum mechanical calculation of chiral anomalies can be extended to trace anomalies关8,9兴. However, in the latter case the details of how to define the path integral is essential since one-loop 共in target space兲 trace anomalies correspond to higher-loop calculations on the worldline, namely the one-loop trace anomaly in D dimensions is given by a D/2⫹1 loop calculation on the worldline. Several regularization schemes have been developed for this purpose: mode regu-larization 共MR兲 关8–10兴, time slicing 共TS兲 关11,12兴, and di-mensional regularization 共DR兲 关13,14兴. The DR regulariza-tion was developed after the results of Ref.关15兴 which dealt with the nonlinear sigma model in the infinite propagation time limit.1 The first objective of this paper is to extend dimensional regularization to include fermionic fields on the worldline and treat supersymmetric nonlinear sigma models. Worldline fermions coupled to gravity give rise to new 共su-perficially兲 divergent Feynman diagrams, other than those associated to the coupling of gravity with the bosonic coor-dinates. Hence one may a priori expect additional counter-terms to arise. In fact, in time slicing, the inclusion of the fermionic fields brings in additional noncovariant counter-terms of order␤2: they are proportional to g␮␯⌫_␮␳␭ ⌫_␯␭␳ if one uses fermions with curved target space indices, or g␮␯␻␮ab␻␯ab if one uses fermions with flat space indices 关12兴. Note that such counterterms only arise at two loops, and thus they do not affect the calculation of the chiral anomalies, but should be included if one wants to check with TS that there are no higher order corrections in ␤ 关17兴. We are going to show that in dimensional regularization no extra counterterms arise. This implies that dimensional

regulariza-*Email address: [email protected]

†_{Email address: [email protected]} ‡_{Email address: [email protected]}

1_{Recently, Kleinert and Chervyakov关16兴 have also analyzed}

non-linear sigma models for finite propagation time, discussing how DR defines products of distributions, and finding results for the Feyn-man rules which agree with those obtained in Ref.关13兴.

tion manifestly preserves supersymmetry. In fact the bosonic part produces a coupling to the scalar curvature with the precise coefficient required by supersymmetry. We describe how to use both flat target space indices and curved ones, for the fermionic fields. Using curved indices will bring in a new set of bosonic ‘‘ghost’’ fields, in the same fashion of Refs. 关8,9兴.

Having at hand a simple and reliable regularization scheme for supersymmetric sigma models, we turn to the worldline formalism. As a warm up, we first compute the trace anomaly for a Dirac fermion in 4 dimensions. We ob-tain the expected result, providing a further test on our ap-plication of the DR scheme. Then we come to the core of our paper: the generalization of the worldline formalism to in-clude spin 1/2 particles coupled to gravity. Many simplifica-tions are known to occur in the worldline path integral for-mulation of quantum field theory共QFT兲, which for this very reason provides an efficient and alternative method for com-puting Feynman diagrams. This method has quite a long his-tory, rooted in Ref. 关18兴. Later it was developed further by viewing it as the particle limit of string theory关19兴, and then discussed directly as the first quantization of point particles 关20,21兴 共see Ref. 关22兴 for a review and a list of references兲. The inclusion of background gravity was presented in Ref. 关2兴 for the case of a scalar particle. Results obtained using string inspired rules with gravity were presented in Refs. 关23,24,3兴.

Here we consider the case of the one-loop effective action for a Dirac fermion in a gravitational background. We de-scribe how to represent it as a worldline path integral. We compute explicitly the one- and two-point correlation func-tions, i.e. the spin 1/2 particle contribution to the graviton tadpole and graviton self-energy. These results are presented for the general case of a massive fermion. In our calculations we use the DR scheme constructed in the previous sections. The other known scheme explicitly developed to include worldline fermions 共time slicing 关12兴兲 can be used as well, but lack of manifest covariance makes its use more compli-cated. It is interesting to note that in the worldline formalism the coupling to gravity can be described entirely in terms of the metric, avoiding the introduction of a vielbein. The fermion-graviton vertices are always linear in the metric field, just like the standard coupling of fermions to gauge fields are linear in the gauge potential. This fact seems to point once more to the unexpected perturbative relations be-tween gravity and gauge theories encoded in the so-called Kawai-Lewellen-Tye 共KLT兲 relations 关25兴, as reviewed in Ref. 关3兴.

The paper is organized as follows. In Sec. II we introduce dimensional regularization applied to the worldline Majorana fermions and to supersymmetric sigma models. We mainly consider antiperiodic boundary conditions 共which break su-persymmetry兲, but also briefly discuss periodic boundary conditions. In Sec. III we apply DR to compute the trace anomaly of a Dirac field in 4 dimensions with quantum me-chanics. Then in Sec. IV we describe the worldline formal-ism with Dirac fields coupled to gravity and compute explic-itly the spin 1/2 particle contribution to the graviton tadpole and graviton self-energy. This is the main section, and the

reader uninterested in the details of the DR regularization scheme may jump directly to it. Section V contains our con-clusions. Conventions and useful formulas are collected in the Appendix. We work with a Euclidean time both on the worldline and in target space. The latter is assumed to have even dimensions D.

II. DIMENSIONAL REGULARIZATION WITH FERMIONS

In this section we describe the dimensional regularization of fermionic path integrals obtained by extending the method presented in Ref. 关13兴 for bosonic models. We shall discuss explicitly path integrals for Majorana fermions on a circle with antiperiodic boundary conditions 共ABC’s兲, as these are the only boundary conditions that will be directly needed in the applications to trace anomalies and effective action cal-culations. Our strategy will be as follows: we first set up the rules of dimensional regularization for fermions following Ref. 关13兴, then we require that a two-loop computation with DR reproduces known results, and precisely those obtained by a path integral with time slicing关12兴 共or equivalently by heat kernel methods关26兴兲. This requirement plays the role of a standard 共in QFT兲 renormalization condition, and fixes once and for all the DR two-loop counterterm due to fermi-ons. Since counterterms are due to ultraviolet effects, the infrared vacuum structure and the related boundary condi-tions on the fields should not matter in their evaluation. Therefore one expects that the same counterterm should ap-ply to fermionic path integral with periodic boundary condi-tions 共PBC’s兲 as well. No higher-loop contributions to the counterterm are expected as the model is super-renormalizable, just as in the purely bosonic case.

Let us consider the path integral quantization of the N ⫽1 supersymmetric model written in terms of Majorana fer-mions with flat target space indices,

Z⫽

冕

DxDaDbDcD␺e⫺S, 共1兲 S⫽1 ␤

冕

0 1 d␶

冉

1 2g␮␯共x兲共x˙ ␮_x˙␯_⫹a␮_a␯_⫹b␮_c␯_兲 ⫹1₂␺a关␺˙a_⫹x˙␮_␻␮a b共x兲␺b兴⫹␤2关V共x兲 ⫹VCT共x兲⫹VCT

⬘

共x兲兴

冊

共2兲 where as usual we have scaled the propagation time␤ out of the action. The propagation time␤ will be considered as the expansion parameter for a perturbative evaluation 共i.e. the loop counting parameter兲. In the action we have included: 共i兲 the bosonic a␮ and fermionic b␮,c␮ ghost fields which ex-ponentiate the nontrivial path integral measure;共ii兲 the coun-terterm VCTwhich arises in the chosen regularization scheme from the bosonic sector and which is fixed in order to pro-duce a quantum Hamiltonian without nonminimal coupling to the scalar curvature R;共iii兲 the additional counterterm VCT

⬘

which may arise from the fermionic sector; and 共iv兲 the

po-tential V⫽18R which is required to have a supersymmetric

quantum Hamiltonian as given by the square of the super-symmetry charge.

The action is classically supersymmetric if all the poten-tial terms multiplied by␤2are set to zero共the ghosts can be trivially eliminated by using their algebraic equations of mo-tion兲. Supersymmetry may be broken by boundary condi-tions, e.g. periodic for the bosons and antiperiodic for the fermions. Here we assume antiperiodic boundary conditions 共ABC’s兲 for the Majorana fermions␺a_(1)⫽⫺␺a_(0). Majo-rana fermions realize the Dirac gamma matrices in a path integral context, and ABC’s compute the trace over the Dirac matrices. For simplicity we consider a target space with even dimensions D, and thus the curved indices␮,␯, . . . and the flat space indices a,b, . . . both run from 1 to D.

One may explicitly compute by time slicing the transition amplitude for going from the background point x0 at time t

⫽0 back to the same point x0 at a later time t⫽␤ using

ABC’s for the Majorana fermions. In the two-loop approxi-mation this calculation gives

Z⬅tr

具

x0兩e⫺␤Hˆ兩x0

典

⫽ 2D/2 共2␲␤兲D/2

冉

1⫺ ␤ 24R⫹O共␤ 2_兲

冊

_共3兲

where the trace on the left-hand side is only over the Dirac matrices, and where

Hˆ⫽⫺1 2ⵜ” ⵜ”⫽⫺ 1 2ⵜ 2_⫹1 8R 共4兲

is the supersymmetric Hamiltonian of theN⫽1 model 关one can normalize the supersymmetric charge as Qˆ⫽(i/

冑

2)ⵜ” , so that Hˆ⫽Qˆ2]. Note that there is an explicit coupling to the scalar curvature in Eq.共4兲, thus one needs to use a potential V⫽1

8R in the action together with the time slicing

counter-terms V_TS⫽⫺1 8R⫹ 1 8g␮␯⌫␮␳␭ ⌫␯␭␳ and VTS

⬘

⫽ 1 16g␮␯␻␮ ab_␻ ␯ab 共see Ref. 关12兴; later on we will derive once more this value of V_TS

⬘

as well兲. Our conventions for the curvature tensors can be found in Sec. 1 of the Appendix.

Now we want to reproduce Eq.共3兲 in dimensional regu-larization with a path integral over Majorana fermions. This will unambiguously fix the additional counterterm V_DR

⬘

due to the fermions. Note that in dimensional regularization the potential V⫽18R cancels exactly with the counterterm VDR ⫽⫺1

8R coming from the bosons 关13兴.

We focus directly on the regularization of the Feynman graphs arising in perturbation theory. To recognize how to dimensionally continue the various Feynman graphs we ex-tend the action in Eq. 共2兲 from 1 to d⫹1 dimensions as follows: S⫽1 ␤

冕

⍀d d_⫹1t

冉

1 2g␮␯共⳵ ␣_x␮_⳵␣x␯_⫹a␮_a␯_⫹b␮_c␯_兲 ⫹1₂␺¯_a␥␣_共⳵␣␺a_⫹⳵␣x␮_␻␮a_b␺b_兲⫹␤2_V DR

⬘

冊

共5兲

where⍀⫽I⫻Rd is the region of integration containing the finite interval I⫽关0,1兴, ␥␣ are the gamma matrices in d⫹1 dimensions satisfying 兵␥␣,␥␤其⫽2␦␣␤, and t␣⬅(␶,t) with ␣⫽0,1, . . . ,d and with boldface indicating vectors in the extra d dimensions. Here we assume that we can first con-tinue to those Euclidean integer dimensions where Majorana fermions can be defined. The Majorana conjugate is defined by␺¯_a⫽␺aTC_⫾with a suitable charge conjugation matrix C_⫾ such that␺¯a␥␣␺b⫽⫺␺¯b␥␣␺a. This can be achieved for ex-ample in 2 dimensions.2It realizes the basic requirement for the Majorana fermions of the N⫽1 supersymmetric model which must have a nonvanishing coupling ␻␮ab␺a␺b⫽ ⫺␻␮ab␺b_␺a_{. The actual details of how to represent C}

⫾and the gamma matrices in d⫹1 dimensions are not important, as the most important thing for the rules which define the DR scheme for fermions is to keep track of how derivatives are going to be contracted in higher dimensions. Apart from the above requirements, no additional Dirac algebra on the gamma matrices ␥␣ in d⫹1 dimensions is needed. With these rules one can recognize from the action共5兲 the propa-gators and vertices in d⫹1 dimensions, and thus rewrite those Feynman diagrams which are ambiguous in one di-mension directly in d⫹1 dimensions.

The bosonic and ghost propagators are as usual and re-ported in Sec. 2 of the Appendix. The fermionic fields with ABC’s on the worldline,␺a(1)⫽⫺␺a(0), can be expanded in half integer modes

␺a_共␶兲⫽

兺

r苸Z⫹1/2␺r

a

e2␲ir␶ 共6兲

and have the following unregulated propagator:

具

␺a_共␶兲␺a_共␴兲

_典

_⫽␤␦ab_⌬ AF共␶⫺␴兲, ⌬AF共␶⫺␴兲⫽

兺

r苸Z⫹1/2 1 2␲ire 2␲ir(␶⫺␴)_{. 共7兲}

Note that the Fourier sum defining the function⌬AF for the antiperiodic fermions is conditionally convergent for ␶⫽␴, and yields

⌬AF共␶⫺␴兲⫽ 1

2⑀共␶⫺␴兲 共8兲

where ⑀(x)⫽␪(x)⫺␪(⫺x) is the sign function 关with the value⑀(0)⫽0, obtained by symmetrically summing the Fou-rier series兴. The function ⌬AF satisfies

⳵␶⌬AF共␶⫺␴兲⫽␦A共␶⫺␴兲 共9兲 where␦A(␶⫺␴) is the Dirac’s delta on functions with anti-periodic boundary conditions

2_{In Euclidean 2 dimensions one can choose␥}1_⫽␴3

, ␥2⫽␴1and C_⫹⫽1. Recall that C_⫾are defined by C_⫾␥␮C_⫾⫺1⫽⫾␥␮T.

␦A共␶⫺␴兲⫽

兺

r苸Z⫹1/2

e2␲ir(␶⫺␴)_{. 共10兲}

The dimensionally regulated propagator obtained by adding a number d of extra infinite coordinates is derived from Eq. 共5兲 and reads

具

␺a_共t兲␺¯b_共s兲

_典

_⫽␤␦ab_⌬

AF共t,s兲 共11兲 where the function

⌬AF共t,s兲⫽⫺i

冕

ddk 共2␲兲d ⫻

兺

r苸Z⫹1/2 2␲r␥0⫹k•␥ᠬ 共2␲r兲2⫹k2 e 2␲ir(␶⫺␴)_eik•(t⫺s) 共12兲 satisfies ␥␣ ⳵ ⳵t␣⌬AF共t,s兲⫽⫺ ⳵ ⳵s␤⌬AF共t,s兲␥ ␤_⫽␦ A共␶⫺␴兲␦d共t⫺s兲 ⬅␦A d_⫹1 共t⫺s兲. 共13兲

The latter are the basic relations which will be used in the application of DR to fermions. They keep track of which derivative can be contracted to which vertex to produce the d⫹1 delta function. This delta function is only to be used in d⫹1 dimensions, as we assume that only in such a situation

the regularization due to the extra dimensions is taking place.3By using partial integration one casts the various loop integrals in a form which can be computed by sending first d→0. At this stage one can use␥0_{⫽1, and no extra factors}

arise from the Dirac algebra in d⫹1 dimensions. This pro-cedure will be exemplified in the subsequent calculations. Having specified how to compute the ambiguous Feynman graphs by continuation to d⫹1 dimensions the DR scheme is now complete.

Now we are ready to perform the two-loop calculation in

the N⫽1 nonlinear sigma model using DR. The bosonic

vertices together with the ghosts, V and VDR give the stan-dard contribution, as for example in Ref. 关2兴. The overall normalization of the fermionic path integral gives the extra factor 2D/2 which equals the number of components of a Dirac fermion in a target space of even dimensions D. This already produces the full expected result in Eq. 共3兲.

Thus the sum of the additional fermion graphs arising

from the cubic vertex contained in ⌬S

⫽兰0

1_{d␶(1/2␤)x˙}␮_␻␮ab␺a_␺b _{and the contribution from the} extra counterterm V_DR

⬘

must vanish at two loops. The cubic vertex arise by evaluating the spin connection at the back-ground point x0 and reads ⌬S3⫽(1/2␤)␻␮ab兰0

1

d␶y˙␮␺a␺b, where y␮denotes the quantum fluctuations around the back-ground point x₀␮ with vanishing boundary conditions at ␶ ⫽0,1. Using Wick contractions 共see Appendix Sec. 2 for the explicit form of the bosonic propagators with vanishing Di-richlet boundary conditions兲 we identify the following non-trivial contribution to

具

e⫺Sint

典

共other graphs vanish trivially兲

共14兲

where dotted lines represent fermions. As usual, we denote with a left/right dot the derivative with respect to the first/ second variable. Using DR this contribution is regulated by

冕

0 1 d␶

冕

0 1 d␴•⌬•共␶,␴兲关⌬_AF共␶,␴兲兴2 →⫺

冕 冕

␣⌬␤共t,s兲tr关␥␣⌬AF共t,s兲␥␤⌬AF共s,t兲兴 共15兲

where _␣⌬_␤(t,s)⬅(⳵/⳵t␣)(⳵/⳵s␤)⌬(t,s) 共note the minus sign obtained in exchanging t and s in the last propagator; it is the usual minus sign arising for fermionic loops兲. We can partially integrate ⳵_␣ without picking boundary terms and obtain 2

冕 冕

⌬_␤共t,s兲tr兵关␥␣⳵_␣⌬AF共t,s兲兴␥␤⌬AF共s,t兲其 ⫽2

冕 冕

⌬␤共t,s兲tr关␦A d⫹1_共t,s兲␥␤_⌬ AF共s,t兲兴 ⫽2

冕

⌬␤共t,t兲tr关␥␤_⌬ AF共t,t兲兴 →2

冕

0 1 d␶⌬•共␶,␶兲⌬AF共0兲⫽0 共16兲

3_{We are not able to show this in full generality, and at this stage}

this rule is taken as an assumption. One way to prove it explicitly would be to compute all integrals arising in perturbation theory at arbitrary d and check the location of the poles.

because ⌬AF(0)⫽

1

2⑀(0)⫽0 共and ␥

0_{⫽1 at d⫽0). As this}

example shows, the Dirac gamma matrices in d⫹1 dimen-sions are just a bookkeeping device to keep track where one can use the Green equation 共13兲. Actually, the vanishing of this graph is achieved already before removing the regular-ization d→0 by using symmetric integration in the momen-tum space representation of⌬AF(t,t).

Thus no contributions arise from the fermions at order␤2, and this fixes

V_DR

⬘

⫽0. 共17兲

This is exactly what one expects to preserve supersymmetry, as the counterterm VDR is exactly canceled by the extra po-tential term V⫽1

8R needed to have the correct coupling to

the scalar curvature in the Hamiltonian 共4兲. Thus dimen-sional regularization without any counterterm preserves the supersymmetry of the classicalN⫽1 action

S⫽1 ␤

冕

0 1 d␶

冉

1 2g␮␯x˙ ␮_x˙␯_⫹1 2␺a共␺˙ a_⫹x˙␮_␻ ␮ab␺b兲

冊

共18兲 since the amount of the curvature coupling brought in by DR is of the exact amount to render the quantum Hamiltonian Hˆ supersymmetric.

To compare with TS, we can compute again the Feynman graph 共14兲, but now using the TS rules. According to Ref. 关12兴 we must use that•_⌬•_{(␶,␴)⫽1⫺␦(␶,␴), and integrate}

the delta function even if it acts on discontinuous functions. The delta function is ineffective as ⑀(0)⫽0, but the rest gives 1 2

具

共⌬S3兲 2

_典

_共TS兲⫽1 2共⫺2兲

冉

1 2␤ ␻␮ab

冊

2 共⫺␤3_兲

冕

0 1 d␶

冕

0 1 d␴1 4 ⫽₁₆␤共␻␮ab兲2_{. 共19兲}

This is canceled by using an extra counterterm V_TS

⬘

⫽ 1

16(␻␮ab)

2 _{which at this order contributes with a term}

⫺␤V_TS

⬘

evaluated at the background point x0. Thus as

ex-pected we recover the counterterm VTS

⬘

found in Ref.关12兴. To summarize, we have proven that DR extended to fer-mions does not require additional counterterms on top of those described in Ref.关13兴. In addition, supersymmetry re-quires that no counterterms should be added at all to the classical sigma model action.

A. Periodic boundary conditions

We present here some comments on the case of Majorana fermions with PBC’s. The mode expansion of␺a(␶) has now only integer modes,

␺a_共␶兲⫽

兺

n_苸Z ␺n

a

e2␲in␶. 共20兲

The zero modes ␺₀a of the free kinetic operator (⳵␶) are treated separately, and the unregulated propagator in the sec-tor of periodic functions orthogonal to the zero mode reads

具

␺

⬘

a_共␶兲␺

_⬘

b_共␴兲

_典

_⫽␤␦ab_⌬ PF共␶⫺␴兲, ⌬PF共␶⫺␴兲⫽

兺

n⫽0 1 2␲ine 2␲in(␶⫺␴)_, 共21兲 where␺

⬘

a_(␶)⫽␺a_(␶)⫺␺ 0 a_{. The function⌬} PFsatisfies ⳵␶⌬PF共␶⫺␴兲⫽␦P共␶⫺␴兲⫺1 共22兲 with␦P(␶⫺␴) the Dirac delta on periodic functions. Its con-tinuum limit can be obtained by summing up the Fourier series and reads „for (␶⫺␴)苸关⫺1,1兴…

⌬PF共␶⫺␴兲⫽ 1

2⑀共␶⫺␴兲⫺共␶⫺␴兲. 共23兲 The dimensionally regulated propagator is instead

具

␺

⬘

a_共t兲␺¯

_⬘

b_共s兲

_典

_⫽␤␦ab_⌬

PF共t,s兲 共24兲 where the function

⌬PF共t,s兲⫽⫺i

冕

ddk 共2␲兲d ⫻

兺

n⫽0 2␲n␥0⫹k•␥ᠬ 共2␲n兲2⫹k2 e 2␲in(␶⫺␴)_eik•(t⫺s) 共25兲 satisfies ␥␣ ⳵ ⳵t␣⌬PF共t,s兲⫽⫺ ⳵ ⳵s␤⌬PF共t,s兲␥ ␤ ⫽关␦P共␶⫺␴兲⫺1兴␦d共t⫺s兲. 共26兲 Even if one uses PBC’s, one does not expect additional coun-terterms in DR, as mentioned earlier. It could be interesting to check in DR the expected ␤ independence of the super-trace which computes the Witten index, i.e. the chiral anomaly. This is given by the path integral with periodic boundary conditions for both bosons and fermions 关4–6兴. The treatment of the bosonic zero modes is known to be somewhat delicate as a total derivative term may appear at higher loops 关2,27兴. However it should be possible to do a manifestly supersymmetric computation using superfields, and one could thus check if these total derivative terms sur-vive in the supersymmetric case and, in case they do, study their meaning.4

4_{In a very recent paper, Kleinert and Chervyakov 关28兴 have}

dis-cussed how to avoid these total derivative terms which appear using naively the bosonic string inspired propagators.

B. Curved indices

It is interesting to consider as well the case of fermions with curved target space indices. This should be equivalent to the case of fermions with flat target space indices: it is just a change of integration variables in the path integral. How-ever it is an useful exercise to work out, as some formulas will become simpler. The classical N⫽1 supersymmetric sigma model is now written as

S⫽1 ␤

冕

0 1 d␶1 2g␮␯共x兲兵x˙ ␮_x˙␯_⫹␺␮_关␺˙␯_⫹x˙␭_⌫ ␭␳ ␯ _共x兲␺␳_兴其. 共27兲 The fermionic term could also be written more compactly using the covariant derivative (D/d␶)␺␯⫽␺˙␯ ⫹x˙␭_⌫

␭␳

␯ _(x)␺␳_{. Note that the action is now expressed in} terms of the metric and Christoffel connection only, and there is no need of introducing the vielbein and spin connec-tion.

The treatment of the bosonic part goes on unchanged. For the fermionic part we can derive the correct path integral measure by taking into account the Jacobian for the change of variables from the free measure with flat indices

D␺a⫽D关ea_␮共x兲␺␮兴⫽Det⫺1关ea_␮共x兲兴D␺␮ ⫽

冉

兿

0⭐␶⬍1

1

冑

det g_␮␯关x共␶兲兴

冊

D␺

␮_{. 共28兲}

Note the inverse functional determinant appearing because of the Grassmann nature of the integration variables. This extra factor arising in the measure can be exponentiated using bosonic ghosts ␣␮(␶) with the same boundary condition of the fermions共ABC’s or PBC’s兲 and it leads to the following extra term in the ghost action:

S_ghextra⫽1 ␤

冕

0 1 d␶1 2g␮␯共x兲␣ ␮_␣␯_{. 共29兲}

One can check that the counterterms of dimensional regular-ization are left unchanged. The full quantum action for the

N⫽1 supersymmetric sigma model now reads

S⫽1 ␤

冕

0 1 d␶1 2g␮␯共x兲兵x˙ ␮_x˙␯_⫹a␮_a␯_⫹b␮_c␯ ⫹␺␮_关␺˙␯_⫹x˙␭_⌫ ␭␳ ␯ _共x兲␺␳_兴⫹␣␮_␣␯_{其 共30兲} and appears in the path integral as

Z⫽

冕

DxDaDbDcD␺D␣e⫺S. 共31兲

It is clear that supersymmetry is not broken by the boundary conditions if one uses PBC’s. Then the effects of the ghosts cancel by themselves: the ghosts have the same boundary conditions and can be eliminated altogether from the path integral

冉

0⭐␶⬍1

兿

冑

det g_␮␯关x共␶兲兴

冊

冉

兿

0⭐␶⬍1 1

冑

det g_␮␯关x共␶兲兴

冊

⫽1. 共32兲 One can recognize that the potential divergences arising in the bosonic x˙x˙ contractions are canceled by the fermionic ␺␺˙ contractions, while the remaining UV ambiguities are treated by dimensional regularization as usual. In this scheme it should be simpler for example to test that the Wit-ten index 共i.e. the gravitational contribution to the chiral anomaly for a spin 1/2 field兲 does not get higher order con-tributions in worldline loops, and is thus␤ independent.

If one uses ABC’s the ghosts have different boundary con-ditions. Hence their cancellation is not complete, and they must be kept in the action.

III. TRACE ANOMALY FOR A SPIN 1Õ2 FIELD IN 4 DIMENSIONS

As a further test on the DR scheme applied to fermions, we compute the trace anomaly for a spin 1/2 fields in 4 dimensions. This anomaly is given by:共i兲 extending the for-mula共3兲 to include the three-loop correction 共order␤2inside the round bracket兲; 共ii兲 setting D⫽4; 共iii兲 picking up the order␤0 term 关8兴; and 共iv兲 including an overall minus sign which takes care of the fermionic nature of the target space loop. The bosonic part has been computed already in DR using Riemann normal coordinates 共see Ref. 关14兴, use ␰ ⫽1

4, and recall our conventions on the scalar curvature

re-ported in Appendix Sec. 1兲. Multiplied by 2D/2 共the addi-tional normalization due the worldline ABC Majorana fermi-ons兲 it reads

Zbos⫽tr

具

x0兩e⫺␤Hˆ兩x0典bos

⫽ 2 D/2 共2␲␤兲D/2

冉

1⫺ ␤ 24R⫹ ␤2 1152R 2_⫹ ␤ 2 720共R␮␯␭␳ 2 ⫺R_␮␯2 _兲⫺ ␤ 2 480ⵜ 2_R⫹O共␤3_兲

冊

_{. 共33兲}

We have now to include the fermionic contributions. On top of Riemann normal coordinates we may use a Fock-Schwinger gauge for the spin connection ␻_␮ab(x0⫹y)

⫽1

2y␯R␯␮ab(x0)⫹ . . . with y␮ the Riemann normal

coordi-nates around the background point x₀␮. Then the leading quartic vertex S4,f⫽(1/4␤)R␮␯ab兰0

1

d␶y␮y˙␯␺a␺b which originates from the spin connection produces the following 3-loop diagram:

共34兲

where all functions ⌬ and ⌬AF are functions of ␶ and␴ in

this precise order 关recall that ⌬AF is antisymmetric, ⌬AF(␶,␴)⫽⫺⌬AF(␴,␶)].

Now we regulate this graph in DR as follows关the second contribution in Eq. 共34兲 does not need regularization and could be directly computed at d⫽0, but we carry it along anyway兴

共35兲

where we have integrated by parts the ␣ derivative in_␣⌬_␤, which then produces a delta function when acting on fermi-ons 共‘‘equations of motion terms’’兲. This delta function is integrated in d⫹1 dimensions and gives a vanishing contri-bution since⌬AF(0)⫽0. The remaining terms are then

com-puted at d→0. Thus

冓

1 2共S4,f兲 2

冔

_⫽⫺ ␤ 2 384R␮␯ab 2 . 共36兲 This fermionic contribution must now be added to the terms inside the round bracket of Eq.共33兲. Setting D⫽4 one rec-ognizes the following anomaly:

Z兩_␤0_⫺term⫽ 1 4␲2

冉

1 288R 2_⫺ 7 1440R␮␯␭␳ 2 _⫺ 1 180R␮␯ 2 ⫺₁₂₀1 ⵜ2_R

冊

_{. 共37兲}

This is the correct trace anomaly for a Dirac fermion in 4 dimensions once we include the minus sign due to the target space fermionic loop.

IV. ONE-LOOP EFFECTIVE ACTION FOR A DIRAC FIELD IN A GRAVITATIONAL BACKGROUND

It is known that, for a wide class of field theories, the one-loop effective action and the relative N-point vertex functions can be computed using one-dimensional path inte-grals 关21,22兴. Two of us have presented in Ref. 关2兴 the ex-tension of this formalism to include a gravitational back-ground, considering the simplest case of a scalar field. The extension of DR to worldline fermions allows us to do the same for a Dirac field. We will get an expression for the effective action from which we derive explicitly the one- and two-point correlation functions, namely the contribution to the tadpole and self-energy of the graviton. We perform this program considering both flat and curved indices for the worldline Majorana fermions. The use of flat indices pro-duces an effective action⌫¯关e_a␮兴 which is naturally a func-tional of the vielbein. The use of curved indices produces instead an effective action⌫关g␮␯兴 which is naturally a func-tional of the metric. Local Lorentz invariance guarantees that

⌫¯关ea␮兴⫽⌫关g␮␯(ea␮)兴. In the following we shall discuss

both cases. As we shall see, the simplest set up is to use curved indices: in this case the sigma model couples linearly to the metric fluctuations h_␮␯⫽g_␮␯⫺␦_␮␯, and the effective

N-point vertices for the metric are obtained by integrating

over the proper time the quantum average of N graviton ver-tex operators.

A. The worldline formalism

Let us consider the one-loop effective action obtained by quantizing a Dirac field ⌿ coupled to gravity through the vielbein ea␮,

S关⌿,⌿¯ ,ea␮兴⫽

冕

dDxe⌿¯共ⵜ” ⫹m兲⌿ 共38兲

where e⫽det ea_␮, ␻_␮ab is the spin connection, and

ⵜ” ⫽␥a_e

a␮ⵜ␮, ⵜ␮⫽⳵␮⫹

1 4␻␮ab␥

a_␥b_{. 共39兲}

The effective action depends on the background vielbein field ea␮ and formally reads as 关e⫺⌫¯[ea␮] ⬅兰D⌿D⌿¯ e⫺S[⌿,⌿¯ ,ea␮]_{⫽Det(ⵜ” ⫹m)兴}

For a Dirac field one does not expect anomalies共the Euclid-ean effective action is real兲 and one can exploit standard arguments to write

⌫¯关ea␮兴⫽⫺log关Det共ⵜ” ⫹m兲Det共⫺ⵜ”⫹m兲兴1/2 ⫽⫺1 2Tr log共⫺ⵜ” 2_⫹m2_兲 ⫽⫺1 2Tr log

冉

⫺ⵜ 2_⫹1 4R⫹m 2

冊

_{. 共41兲}

In this formula we recognize the logarithm of an operator which up the mass term is proportional to the supersymmet-ric Hamiltonian 共4兲. Thus we can immediately write down a path integral representation for the effective action in terms of a proper time as ⌫¯关ea␮兴⫽ 1 2

冕

0 ⬁dT T

冖

PBCDx ␮

冖

ABC D␺ae⫺S[x␮,␺a;ea␮] 共42兲 where5 S关x␮,␺a;ea␮兴⫽

冕

0 1 d␶

冉

1 4Tg␮␯共x兲x˙ ␮_x˙␯_⫹ 1 4T␺a␺˙ a ⫹_4T1 x˙␮␻␮ab共x兲␺a␺b⫹Tm2

冊

. 共43兲 The subscripts PBC and ABC remind of the boundary con-dition at␶⫽0,1, periodic for the bosonic coordinates x␮(␶) and antiperiodic for the fermionic ones␺a(␶): these bound-ary conditions have to be imposed to obtain the trace in Eq. 共41兲. We have used a rescaled proper time T⫽␤/2 with re-spect to the previous sections to agree with standard normal-izations used in the worldline formalism 关22兴. We have not added any counterterm since we are going to use dimen-sional regularization to compute the path integral.6 Of course, the covariant measure in Eq.共42兲 contains the ghost fields Dx␮_⫽Dx␮

兿

0⭐␶⬍1

冑

det g_␮␯关x共␶兲兴 ⫽Dx␮

冖

PBC Da␮Db␮Dc␮e⫺Sgh[x,a,b,c] 共44兲 where Sgh关x,a,b,c兴⫽

冕

0 1 d␶ 1 4Tg␮␯共x兲共a ␮_a␯_⫹b␮_c␯_{兲. 共45兲}

One may also compute the effective action directly as a func-tional of the metric. This is achieved by using the sigma model written in terms of the Majorana fermions with curved indices. The corresponding formula is

⌫关g␮␯兴⫽ 1 2

冕

0 ⬁dT T

冖

PBCDx ␮

冖

ABCD␺ ␮_e⫺S[x␮,␺␮;g_␮␯] 共46兲 with S关x␮,␺␮;g_␮␯兴⫽

冕

0 1 d␶

冉

1 4Tg␮␯共x兲共x˙ ␮_x˙␯_⫹␺␮_␺˙␯ ⫹␺␮_⌫ ␭␳ ␯ _共x兲x˙␭_␺␳_兲⫹Tm2

冊

_{. 共47兲}

Note that the covariant fermionic measure now contains the new bosonic ghost␣␮

D␺␮_⫽D␺␮

兿

0⭐␶⬍1 1

冑

det g_␮␯关x共␶兲兴 ⫽D␺␮

冖

ABC D␣␮e⫺Sgh extra_[x,␣] 共48兲 with Sgh extra_{关x,␣兴⫽}

冕

0 1 d␶ 1 4Tg␮␯共x兲␣ ␮_␣␯_{. 共49兲} The fermionic term in the action 共47兲 may be written using the covariant derivative as g␮␯␺␮(D/d␶)␺␯, making mani-fest its geometrical meaning. However, one can write the Christoffel connection directly in terms of the metric and, because of the Grassmannian nature of the fields ␺␮, the action simplifies to S关x␮,␺␮;g_␮␯兴⫽

冕

0 1 d␶

冉

1 4Tg␮␯共x兲共x˙ ␮_x˙␯_⫹␺␮_␺˙␯_兲 ⫺_4T1 ⳵␮g_␯␭共x兲␺␮␺␯x˙␭⫹Tm2

冊

共50兲 which shows that there is only a linear coupling to the back-ground g_␮␯(x). To summarize, we have two options for rep-resenting the effective action in the worldline formalism, and we will consider both of them.

The next step is to discuss how to treat the boundary conditions. Due to the translational invariance of the result-ing propagators, we adopt the ‘‘strresult-ing inspired’’ option: one expands the coordinate fields with periodic boundary condi-tions into Fourier modes and then separates the zero mode x₀␮⫽兰₀1d␶x␮(␶) from the quantum fluctuations y␮(␶) ⫽x␮_(␶)⫺x

0

␮_{. The latter have an invertible kinetic term and} the integration over the constants zero mode x₀␮is performed

5_{Presumably, this final action can be obtained also by gauge fixing}

the locally supersymmetric formulation of the spinning particle ac-tion 关29,30兴, at least in the massless case, as the corresponding ghosts decouple from the background geometry and can be ignored.

6_{Let us recall that other regularization schemes共such as time}

separately. For the alternative option of using Dirichlet boundary conditions, see a discussion in Ref.关2兴. Other op-tions for treating the zero modes can be found in Refs. 关31,22兴.

These subtleties do not arise for the anticommuting vari-ables␺aas the boundary conditions are now antiperiodic and the kinetic term has no zero mode. All these propagators are collected in Sec. 3 of the Appendix.

For later convenience, it is useful to introduce the follow-ing notations: ⌫¯(x₁, . . . ,x_N) a₁␮₁•••a_N␮_N ⫽ ␦ N_⌫¯ ␦ea₁␮₁共x1兲•••␦ea_N␮_N共xN兲

冏

e_a␮⫽␦_a␮ , 共51兲 ⌫(x₁, . . . ,x_N) ␮1␯1•••␮N␯N⫽ ␦ N_⌫ ␦g_␮ 1␯1共x1兲•••␦g␮N␯N(xN)

冏

_g ␮␯⫽␦␮␯ 共52兲 and the corresponding Fourier transform for the vielbein ver-tex functions: ⌫ន( p₁, . . . , p_N) a₁␮₁•••a_N␮_N ⫽共2␲兲D_␦D_共p 1⫹ . . . ⫹pN兲⌫¯( p₁, . . . , p_N) a₁␮₁•••a_N␮_N ⫽

冕

dx₁. . . dx_Nei p₁x₁⫹ . . . ⫹ip_Nx_N⌫¯ (x₁, . . . ,x_N) a₁␮₁•••a_N␮_N 共53兲 plus a similar one for the metric vertex functions. The cor-relation functions obtained by varying the vielbein are sym-metric under the interchange of indices belonging to the same couple as a consequence of local Lorentz invariance. Notice also that after restricting these vertices to flat space there is no intrinsic difference between curved and flat indi-ces. For N⫽1,2, and using ⌫¯关ea␳兴⫽⌫关g␮␯(ea␳)兴 together with the relation g_␮␯⫽␦_abea

␮eb␯, one finds ⌫(x)␮␯⫽ 1 2⌫¯(x) ␮␯_{, 共54兲} ⌫_{(x,y )}␮1␯1␮2␯2⫽1 4⌫¯(x,y ) ␮1␯1␮2␯2⫺1 4⌫¯(x) ␮1␮2␦␯₁␯₂␦D_{共x⫺y兲 共55兲} where we indicate with underline and overline a normalized symmetrization.

Following a standard technique, one can obtain the vertex functions directly in momentum space 关22兴. Let us describe it for the effective action⌫¯关ea␮兴. One considers ⌫¯关ea␮兴 as a power series in ca␮⬅ea␮⫺␦a␮ 共note that this definition induces a relative expression for the metric: g_␮␯⫽␦_␮␯ ⫹c␮␯⫹c␯␮⫹ca␮c_␯

a

, where c_␮␯⫽ca␯␦_␮ a

), takes the cN term as a sum of N plane waves of given polarizations共our polar-ization tensors include the gravitational coupling constant兲

ca␮共x兲⫽

兺

i⫽1 N ␧a␮ (i) ei pi•x_, 共56兲

and then picks up the terms linear in each ␧_a(i)_␮: this gives directly the N-point function in momentum space,

⌫ន_{( p} 1, . . . , pN) ␧1•••␧N ⬅␧ a₁␮₁ (1) _•••␧ a_N␮_N (N) _⌫ន ( p₁, . . . , p_N) a₁␮₁•••a_N␮_N 共57兲 关the tilde symbol can be dropped by removing the momen-tum delta functions as in Eq. 共53兲兴.

In the following sections we are going to compute the one- and two-point correlation functions. We will employ the worldline ‘‘string inspired’’ propagators together with dimen-sional regularization on the worldline 共and in target space兲.

B. One- and two-point functions from⌫¯†ea µ‡

The one-point vertex function can be depicted by the Feynman diagram of Fig. 1 where the external line refers to the vielbein. It gives the Dirac particle contribution to the cosmological constant. The recipe just outlined tells that the term in the effective action linear in ca␮, and with ca␮ ex-pressed as a single plane wave, produces

⌫¯˜( p)␧ ⫽ 1 2

冕

0 ⬁dT T e ⫺m2_T 2D/2 共4␲T兲D/2

冕

d D_x 0

冉

⫺ 1 4T

冊

⫻

冕

0 1

d␶兵

具

2␧(␮␯)共y˙␮y˙␯⫹a␮a␯⫹b␮c␯兲ei p•(x0⫹y)

典

⫹

具

y˙␮␻_␮ab(1) 共x0⫹y兲

典具

␺a␺b

典

其, 共58兲

where the superscript on␻_␮ab(1) denotes the part linear in␧a␮, and round brackets around indices denote symmetrization normalized to 1.

It can be immediately noted that the contribution of the spin connection term vanishes, being proportional to ␻␮ab␦ab_⌬

AF(0)⫽0. Therefore everything proceeds as in the scalar field case 关2兴, and the one-point function reads

⌫¯(0)␮␯⫽2 D/2␦ ␮␯ 2 共m2_兲D/2 共4␲兲D/2⌫

冉

⫺ D 2

冊

. 共59兲

Clearly it diverges for even target space dimension D and renormalization is needed.

Let us now discuss the two-point vertex function. We set ca␮共x兲⫽␧a␮ (1) ei p1•x⫹␧ a␮ (2) ei p2•x_. 共60兲

One sees that there are two kinds of contributions, illustrated by the Feynman graphs in Figs. 2 and 3, which we denote as ⌬1⌫¯␮␯␣␤and⌬2⌫¯␮␯␣␤, respectively.

In the first one there is just one vertex. It is simple to compute it, being quite similar to the tadpole. It reads

⌬1⌫ន( p₁, p₂) ␧1␧2 ⫽1 2

冕

0 ⬁dT T e ⫺m2_T 2D/2 共4␲T兲D/2

冕

d D_x 0

冉

⫺ 1 4T

冊

⫻

冕

0 1 d␶兵

具

ca␮c␯ a_共y˙␮ y˙␯⫹a␮a␯⫹b␮c␯兲

典

⫹

具

y˙␮␻_␮ab(2) 共x0⫹y兲

典具

␺a␺b

典

其兩m.l. 共61兲

where␻_␮ab(2) is the part of the spin connection quadratic in the ca␮ field; the prescription m.l.共multilinear兲 refers to the two different polarization tensors. The contribution from the spin connection term vanishes for the same reason as before. We are then left with the bosonic contribution, which gives

⌬1⌫¯( p,␮␯␣␤⫺p)⫽ 2D/2 4 共␦ ␮␣_␦␯␤_⫹␦␮␤_␦␯␣_兲共m 2_兲D/2 共4␲兲D/2⌫

冉

⫺ D 2

冊

共62兲 where, according to the notation 共53兲, we have factored out (2␲)D␦D_{( p}

1⫹p2) and used p⫽p1⫽⫺p2.

The two-vertex graph of Fig. 3 produces

⌬2⌫ន( p₁, p₂) ␧1␧2 ⫽1 2

冕

0 ⬁dT T e ⫺m2_T 2D/2 共4␲T兲D/2 ⫻

冕

dDx0

冓

1 2

冋

冕

0 1 d␶

冉

1 2Tc(␮␯)共y˙ ␮_y˙␯_⫹a␮_a␯ ⫹b␮_c␯_兲⫹ 1 4Ty˙ ␮_␻ ␮ab (1) _␺a_␺b

冊册

2

冔

冏

m.l. .

Three kinds of contributions are included in the previous expression:共i兲 the square of the bosonic part which yields a term proportional to the contribution of a scalar field 共non-minimally coupled with ␰⫽1/4, already computed in Ref. 关2兴兲; 共ii兲 the mixed terms of the product which are zero, again being proportional to␻␮ab␦ab⌬AF(0); 共iii兲 the square of the fermionic term which contains

␻␮ab(1)共x0⫹y兲⫽

兺

i⫽1 2 共⫺ip␮␧[ab] (i) _⫹i␧ ␮[a (i) _p b] ⫺ip[a␧b]␮

(i) _兲ei p_i•(x₀⫹y) _共63兲

where square brackets around indices denote antisymmetri-zation normalized to 1. This third term produces the follow-ing contribution:

冕

0 ⬁ dT 32T3e ⫺m2_T 2D/2 共4␲T兲D/2共⫺ip␮␧[ab] (1) _⫹i␧ ␮[a (1)_p b] ⫺ip[a␧b]␮ (1)_兲共⫺ip ␳␧[cd] (2) _⫹i␧ ␳[c (2) pd]⫺ip[c␧d]␳ (2)_兲

冕

0 1 d␶ ⫻

冕

0 1

d␴

具

ei p•yy˙␮␺a␺b共␶兲e⫺ip•yy˙␳␺c␺d共␴兲

典

. 共64兲 After performing Wick contractions, the second line of this expression becomes 共␦ac_␦bd_⫺␦ad_␦bc_兲共2T兲3

冕

0 1 d␶

冕

0 1 d␴兵␦␮␳ •⌬•共␶⫺␴兲 ⫹2Tp␮_p␳ •_⌬2_{共␶⫺␴兲其e}⫺2Tp2⌬₀(␶⫺␴)_⌬ AF 2 _{共␶⫺␴兲, 共65兲}

where ⌬0(␶⫺␴)⫽⌬(␶⫺␴)⫺⌬(0), and needs worldline

regularization. Following the rules of dimensional regular-ization we write the last line of the above expression as we would have done starting from the action in 1⫹d dimen-sions,

冕 冕

兵␦␮␳ ␣⌬␤共t⫺s兲⫹2Tp␮p␳␣⌬共t⫺s兲␤⌬共t⫺s兲其 ⫻e⫺2Tp2_⌬ 0(t⫺s)_tr关␥␣⌬ AF共t⫺s兲␥␤⌬AF共t⫺s兲兴 共66兲 and perform an integration by parts on the ␣ index of the first term, as already explained in Eqs.共16兲 and 共35兲, to get the following result:

⫺1₂T p2

冉

␦␮␳⫺p ␮_p␳ p2

冊

冕

0 1 d␶

冉

␶⫺1 2

冊

2 e⫺Tp2(␶⫺␶2). 共67兲 The remaining worldline integral can be computed as de-scribed in Sec. 4 of the Appendix. Using the result into Eq. 共64兲 gives ␧_␮␯(1)_␧ ␣␤ (2)1 8 2D/2 共4␲兲D/2⌫

冉

1⫺ D 2

冊

p 2_关共P2_兲D/2⫺1 ⫺共m2_兲D/2⫺1_兴S 2 ␮␯␣␤ _共68兲

where P2 and S2are defined below. Collecting all terms, we

find for the two-vertex part of the self-energy

⌬2⌫¯( p,⫺p)⫽ 2D/2 2共4␲兲D/2

再

⌫

冉

⫺ D 2

冊

兵共m 2_兲D/2_共R 1⫺R2兲 ⫹关共P2_兲D/2_⫺共m2_兲D/2_兴共S 1⫹S2兲其 ⫹1₄⌫

冉

1⫺D 2

冊

p 2_共P2_兲D/2⫺1_S 2

冎

. 共69兲

FIG. 2. One-vertex graph for graviton self-energy.

Consequently, the full graviton self-energy obtained by sum-ming Eqs.共62兲 and 共69兲 reads

⌫¯( p,_⫺p)⫽ 2D/2 2共4␲兲D/2

再

⌫

冉

⫺ D 2

冊冋

共m 2_兲D/2

冉

_R 1⫺ 1 2R2⫺S1⫺S2

冊

⫹共P2_兲D/2_共S 1⫹S2兲

册

⫹ 1 4⌫

冉

1⫺ D 2

冊

⫻p2_共P2_兲D/2⫺1_S 2

冎

共70兲

where, as in Ref.关2兴, we have suppressed tensor indices, and used the following basis of dimensionless tensors with the required symmetry properties:

R₁␮␯␣␤⫽␦␮␯␦␣␤, R₂␮␯␣␤⫽␦␮␣␦␯␤⫹␦␮␤␦␯␣, R₃␮␯␣␤⫽ 1 p2共␦␮␣p␯p␤⫹␦␯␣p␮p␤⫹␦␮␤p␯p␣ ⫹␦␯␤_p␮_p␣_兲, R₄␮␯␣␤⫽ 1 p2共␦ ␮␯_p␣_p␤_⫹␦␣␤_p␮_p␯_兲, R₅␮␯␣␤⫽ 1 p4p ␮_p␯_p␣_p␤_{. 共71兲}

For simplicity we have introduced the manifestly transverse combinations S₁␮␯␣␤⫽R1⫺R4⫹R5⫽

冉

␦␮␯⫺ p␮p␯ p2

冊冉

␦ ␣␤_⫺p␣p␤ p2

冊

, 共72兲 S2␮␯␣␤⫽R2⫺R3⫹2R5⫽2

冉

␦␮គ␣¯⫺ p␮គp␣¯ p2

冊冉

␦ ␯គ␤¯_⫺p␯គp␤¯ p2

冊

共73兲 and defined 共P2_兲x_⫽

冕

0 1 d␶关m2⫹p2共␶⫺␶2兲兴x. 共74兲 Further details may be found in Sec. 4 of the Appendix.

The final results for the one- and two-point functions, Eqs. 共59兲 and 共70兲, satisfy the gravitational Ward identities 共see Appendix Sec. 5兲. Of course, one may now extract the divergent part and renormalize these functions in the chosen spacetime dimensions.7

C. One- and two-point functions from⌫†gµ␯‡

In this section we describe the calculation of the one- and two-point functions employing curved indices for the world-line Majorana fermions. As already explained in Secs. II B and IV A, the total action including the ghost fields is given by S⫽

冕

0 1 d␶

再

1 4Tg␮␯共x兲共x˙ ␮_x˙␯_⫹a␮_a␯_⫹b␮_c␯_⫹␺␮_␺˙␯_⫹␣␮_␣␯_兲 ⫺_4T1 ⳵␮g_␯␭共x兲␺␮␺␯x˙␭⫹Tm2

冎

. 共75兲 Clearly, there are no vertices with two or more gravitons in this picture. Using

h_␮␯共x兲⬅g_␮␯共x兲⫺␦_␮␯⫽

兺

i⫽1 N

⑀␮␯(i)ei pi•x 共76兲 with the gravitational coupling constant included into the polarization tensors, one gets the following general expres-sion for the N-point effective vertices:

1 2

冕

0 ⬁dT T e ⫺m2_T 2D/2 共4␲T兲D/2

冉

⫺ 1 4T

冊

N

冓

兿

i_⫽1 N

冕

0 1

d␶iV(i)共␶i兲

冔

共77兲 where the graviton vertex operator V(i)(␶i) is given by

V(i)共␶i兲⫽⑀_␮␯(i)共x˙␮x˙␯⫹a␮a␯⫹b␮c␯⫹␺␮␺˙␯⫹␣␮␣␯ ⫺ipi␭␺␭x˙␮␺␯兲共␶i兲e

i p_i•x(␶_i)_{. 共78兲}

The explicit calculations of the one- and two-point vertex functions—depicted in Figs. 4 and 5, respectively—give the same results previously obtained from the coupling to the vielbein关after using relations 共54兲 and 共55兲兴. Note that exter-nal lines in Figs. 4 and 5 now refer to metric fluctuations. Let us describe briefly these calculations.

In the one-point function the connection term关i.e. the last term inside the round brackets of the vertex operator 共78兲兴 does not contribute, and the remaining terms lead to the same worldline integral obtained previously,

7_{If one is interested in odd dimensions, then there is no divergence}

at one loop, but the formulas should be modified by substituting 2D/2_→2[D/2]_{to account for the correct number of components of a}

Dirac spinor. Here关D/2兴 denotes the integer part of D/2.

FIG. 4. Graviton tadpole.

冕

0 1 d␶共•_⌬•_⫹⌬ gh⫺⌬AF • _⫺⌬ Agh兲共0兲 ⫽

冕

0 1 d␶共•⌬•⫹⌬_gh兲共0兲⫽1. 共79兲 In fact, the propagator of the extra ghost field

具

␣␮_共␶兲␣␯_共␴兲

_典

_⫽2T␦␮␯_⌬

Agh共␶⫺␴兲 共80兲 where

⌬Agh共␶⫺␴兲⫽•⌬AF共␶⫺␴兲⫽␦A共␶⫺␴兲 共81兲 cancels with ⌬_AF• (␶⫺␴)⫽⫺•⌬AF(␶⫺␴) due to the fermi-onic propagator. This exemplifies the effect of the new ghost ␣␮_{which cancels a contraction arising from the␺}␮_␺˙␯_term. The final answer is

⌫(0)␮␯⫽2 D/2␦ ␮␯ 4 共m2_兲D/2 共4␲兲D/2⌫

冉

⫺ D 2

冊

. 共82兲

As one might have expected, this result is ⫺2D/2times the contribution of a scalar field关2兴: the minus sign is the usual one due to a fermionic loop, while 2D/2 is the number of degrees of freedom of a Dirac fermion in even D dimensions. Let us now look at the two-point function. It corresponds to the single diagram of Fig. 5, as in this scheme all vertices are linear in the graviton field. Again one may note that the ␣␮ _{ghosts cancel all Wick contractions arising from the␺␺}˙ term of the vertex operators 共notice that ⌬AF⌬AF

• _⫽

⫺⌬AF␦A⫽0). Thus only two nonvanishing contributions survive: one from the square of the kinetic term of the bosonic sector 关i.e. ⬃(x˙2⫹a2⫹bc)2]; the other, transverse, from the square of the connection term. The final result is

⌫( p,⫺p)⫽ 2D/2 8共4␲兲D/2

再

⌫

冉

⫺ D 2

冊

关共m 2_兲D/2_共R 1⫺R2⫺S1⫺S2兲 ⫹共P2_兲D/2_共S 1⫹S2兲兴⫹ 1 4⌫

冉

1⫺ D 2

冊

⫻p2_共P2_兲D/2⫺1_S 2

冎

. 共83兲

This expression satisfies the expected gravitational Ward identities共for details see Sec. 5 of the Appendix兲. The gravi-ton self-energy due to a massless fermion has been already computed in Ref.关32兴, and agrees with the massless limit of this general result.8

V. CONCLUSIONS

We have extended the worldline formalism to include fer-mionic fields coupled to gravity. To achieve this task we have found it useful to study dimensional regularization on super-symmetric worldline sigma models. We have shown that di-mensional regularization preserves worldline supersymmetry in that no counterterms need to be added to the classical action to maintain supersymmetry. This is in contrast to the time slicing regularization scheme, previously used for su-persymmetric sigma models, which required specific coun-terterms to restore supersymmetry. Of course, final physical results are independent of the regularization scheme adopted. We have applied this set up to describe quantum properties of a Dirac fermion coupled to gravity. Then, we have de-scribed the one-loop effective action for a Dirac fermion coupled to gravity in the worldline formalism, and computed the corresponding one- and two-point functions, namely the one-loop fermionic contribution to the cosmological constant and graviton self-energy. We have seen that one can use a formulation either in terms of the vielbein or in terms of the metric, the latter being much simpler as the coupling to grav-ity is linear 共and it avoids the introduction of the local Lor-entz symmetry related to a choice of the vielbein兲. The com-putations are rather simple and demonstrate the efficiency of the worldline formalism in computing Feynman graphs even in the presence of gravitational fields. Our conclusion is that one can be confident and address more complicated pro-cesses using the worldline method. In particular, mixed photon-graviton amplitudes are under study关35兴.

ACKNOWLEDGMENTS

We would like to thank Christian Schubert for useful dis-cussions and comments. We also thank the EC Commission for financial support via the FP5 Grant HPRN-CT-2002-00325.

APPENDIX

1. Covariant derivatives and curvature tensors

The covariant derivative for a vector with curved indices is ⵜ_␮V␯⫽⳵_␮V␯⫹⌫_␮␭␯ V␭, where ⌫_␮␭␯ ⫽1

2g␯␳(⳵␮g␭␳⫹⳵␭g␮␳ ⫺⳵␳g␮␭) is the usual Christoffel connection. The corre-sponding curvatures are defined by

关ⵜ␮,ⵜ␯兴V␭⫽R␮␯␭␳共⌫兲V␳, R␮␯⫽R␭␮␭␯共⌫兲, R⫽R␮_␮⬎0 on spheres. 共A1兲 The covariant derivative of a vector with flat indices is ⵜ␮Va⫽⳵␮Va⫹␻␮abVb, where ␻␮ab is the spin connection satisfying the ‘‘vielbein postulate’’ⵜ_␮(⌫,␻)ea_␯⫽0. The cor-responding curvature is

关ⵜ␮,ⵜ␯兴Va⫽R␮␯ab共␻兲Vb. 共A2兲 These curvatures are related by

R_␮␯␭_␳共⌫兲⫽R_␮␯a

b共␻兲e␭aeb␳. 共A3兲

8_{A similar result for a massive scalar field with minimal coupling}

to the scalar curvature can be found in Ref.关33兴 and agrees with the worldline result关2兴. A calculation with standard Feynman rules for the scalar has been recently reported again in Ref.关34兴. It may be noticed how the worldline computation produces simpler and more compact expressions.

2. Propagators for bosons and related ghosts with vanishing Dirichlet boundary conditions

For quantum fields that vanish at ␶⫽0,1, we have the following propagators:

具

y␮共␶兲y␯共␴兲

典

⫽⫺␤␦␮␯⌬共␶,␴兲,

具

a␮共␶兲a␯共␴兲

典

⫽␤␦␮␯⌬gh共␶,␴兲,

具

b␮共␶兲c␯共␴兲

典

⫽⫺2␤␦␮␯⌬gh共␶,␴兲

共A4兲 with Green functions⌬ and ⌬gh satisfying vanishing Dirich-let boundary conditions

⌬共␶,␴兲⫽

兺

m⫽1 ⬁

冉

⫺_␲22_m2sin共␲m␶兲sin共␲m␴兲

冊

⫽共␶⫺1兲␴␪共␶⫺␴兲⫹共␴⫺1兲␶␪共␴⫺␶兲, ⌬gh共␶,␴兲⫽

兺

m⫽1 ⬁ 2 sin共␲m␶兲sin共␲m␴兲 ⫽••_{⌬共␶,␴兲⫽␦共␶,␴兲 共A5兲}

where␪(␶⫺␴) is the standard step function and ␦(␶,␴) is the Dirac’s delta function which vanishes at the boundaries ␶,␴⫽0,1. These functions are not translationally invariant.

Their extensions to d⫹1 dimensions read

⌬共t,s兲⫽

冕

d d_k 共2␲兲d ⫻

兺

m_⫽1 ⬁ _⫺2

共␲m兲2⫹k2sin共␲m␶兲sin共␲m␴兲e ik•(t⫺s)_, 共A6兲 ⌬gh共t,s兲⫽

冕

dd_k 共2␲兲d _m

兺

_⫽1 ⬁

2 sin共␲m␶兲sin共␲m␴兲eik•(t⫺s) ⫽␦共␶,␴兲␦d共t⫺s兲⫽␦d⫹1共t,s兲. 共A7兲 Note that the function ⌬(t,s) satisfies the relation 共Green’s equation兲

⳵␣_{⳵␣⌬共t,s兲⫽⌬}

gh共t,s兲⫽␦d⫹1共t,s兲. 共A8兲 The d→0 limits of these propagators reproduce the unregu-lated expressions.

3. The ‘‘string inspired’’ propagators

The propagators we used in the worldline formalism are the ‘‘string inspired’’ ones. More specifically, on the circle the free kinetic term for x␮ is proportional to ⳵_␶2 and has a zero mode. Thus one splits

x␮共␶兲⫽x0␮⫹y␮共␶兲, x₀␮⫽

冕

0 1 d␶x␮共␶兲, y␮共␶兲⫽

兺

n⫽0 y_n␮e2␲in␶ 共A9兲 and the path integration measure becomes

Dx⫽ 1 共4␲T兲D/2d

D_x

0D y . 共A10兲

The kinetic term for the quantum bosonic fields y␮is invert-ible and the corresponding free path integral is normalized to unity,

冕

D y e⫺兰0 1

d␶(1/4T)y˙2_{⫽1. 共A11兲} The value of the free fermionic path integral defines implic-itly its measure. Using flat indices it reads

冕

ABC

D␺ae⫺兰0 1

d␶(1/2T)␺_a␺˙a_{⫽tr共1兲⫽2}D/2_{. 共A12兲}

The propagators for the free fields are

具

y␮共␶兲y␯共␴兲

典

⫽⫺2T␦␮␯⌬共␶⫺␴兲,

具

a␮共␶兲a␯共␴兲

典

⫽2T␦␮␯⌬_gh共␶⫺␴兲,

具

b␮共␶兲c␯共␴兲

典

⫽⫺4T␦␮␯⌬gh共␶⫺␴兲,

具

␺a_共␶兲␺b_共␴兲

_典

_⫽2T␦ab_⌬ AF共␶⫺␴兲, 共A13兲 where⌬, ⌬_gh and⌬_AF are given by

⌬共␶⫺␴兲⫽⫺

兺

n⫽0 1 4␲2n2e 2␲in(␶⫺␴) ⫽1₂兩␶⫺␴兩⫺₂1共␶⫺␴兲2_⫺ 1 12, ⌬gh共␶⫺␴兲⫽

兺

n⫽⫺⬁ ⬁ e2␲in(␶⫺␴) ⫽␦P共␶⫺␴兲, 共A14兲 ⌬AF共␶⫺␴兲⫽

兺

r苸Z⫹1/2 1 2␲ire 2␲ir(␶⫺␴) ⫽1₂⑀共␶⫺␴兲 and satisfy ••_⌬⫽⌬

gh⫺1⫽␦P⫺1,•⌬AF⫽␦A,where␦P and␦A are the Dirac delta functions on the space of periodic and antiperiodic functions on 关0,1兴, respectively. All these free