ELSEWIER

25 May 1995

Physics Letters B 351 (1995) 200-205

PHYSICS LETTERS B

Yukawa couplings for the spinning particle and the world-line fornialism

Myriam Mondrag6n a~1, Lukas Nellen a*2, Michael G. Schmidt a,3, Christian Schubert b,4 a Institut fiir Theoretische Physik, Universittit Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany

b Humboldt Universitiit zu Berlin, Institut fiir Physik, Invalidenstr. 110, D-10115 Berlin, Germany

Received 28 February 1995

Editor: P.V. Landshoff

Abstract

We construct the world-line action for a Dirac particle coupled to a classical scalar or pseudo-scalar background field. This action can be used to compute loop diagrams and the effective action in the Yukawa model using the world-line path-integral formalism for spinning particles.

In the framework of the Bern-Kosower formal- ism [l-3], derived from string theory, it is possible to reproduce results of tree and one-loop field theory calculations in a compact and elegant form which has its origin in the simplicity of the string theory pertur- bation expansion. Many of the results for one-loop calculations can also be understood through world- line path integrals [4,5]. This approach to quantum field theory can be generalised to multi-loop calcula- tions [ 6,7] _ For earlier applications of world-line path integrals to quantum field theory see also [8-l 11.

In the following we construct the action for the Yukawa coupling for the spinning particle in Euclidean space. While the original Bern-Kosower approach works for Yukawa couplings [ 121 the correct form of the world-line action was so far unknown, even though guesses existed before [4]. Our starting point

1 E-mail address: Myriam.Mondragon@urz.uni-heidelbergde.

2 E-mail address: Lukas.Nellen@urz.uni-heidelberg.de. 3 E-mail address: k22@vm.urz.uni-heidelberg.de.

4 E-mail address: schubert@qft2.physik.hu-berlin.de.

is the well-known world-line action for a massless, spinning particle coupled to a gauge field [ 13,141. By dimensional reduction of the gauge coupling we obtain the form of the Yukawa coupling. For the case of a constant background field we recover the action for the massive, spinning particle as a special case. It turns out that we can generalise this to include pseudo-scalar couplings as well as scalar couplings.

We present some applications to the calculation of one-loop amplitude both to reproduce the result of Feynman diagram calculations and to get a variant of the heat-kernel expansion for the one-loop effective action.

Our starting point is the first-quantized description of a Dirac particle [ 15,13,14], given by a supersym- metric action in one dimension. Such an action can be formulated in a compact way using superfield nota- tion. Furthermore, this notation ensures the supersym- metry of the action, an important point since we want to add new couplings without loosing supersymmetry.

Our interest lies in the application of the world- line formalism to loop calculations with external scalar

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M. Mondragbn et al. /Physics Letters B 351 (1995) 200-205 201

fields. For this reason we do not have to worry about In the process of dimensional reduction we single the problem of boundary conditions for the fermions out the fields Xs and As. Furthermore, we notice that

like in the derivation of the Dirac propagator from the X5 only appears in the combination DXs. Therefore

world-line action [ 16,171. we introduce the following convenient definitions:

In the superfield formulation [ 151, the world-line parameter r is supplemented by an anti-commuting Grassmann parameter 6 to form a two-dimensional superspace. In this formulation, the free spinning par-

ticle is described using world-line superfields with a space-time vector index

8 = A’12DBXs, @ = A’. (7)

With these definitions the world-line action for a spin-

ning particle with a Yukawa coupling is

XP(7,8) = xc”(r) + &L?@“(r), (1)

where x is a normal commuting number, and 8 is a Grassmann variable.

To be able to write down a reparametrisation invari- ant action we need to introduce the super-einbein [ 1.51

Sy=So+; drdeA’12 J x (A-%D,X + 2A-‘/2ih8@ (X)) _ (8)

To rewrite this expression in components we expand the superfield 8, which is fermionic, and the scalar field @ as

A=e+i?&X. (2)

In curved superspace, two independent derivatives ex- ist:

@(X> = @(x) + e&fy”a,@(x>. (9)

(This expansion defines xs for the rest of this letter, not Eq. ( 1). We apologise for the confusion.) In com- ponent notation, the action (8) is

Using these ingredients, the world-line action for a free, spinning particle is

So = ; J drd0 R’12D,X. D@X. (4)

In components, this reads

(5)

The coupling of the spinning particle to a Yang- Mills field is well-known [ 131:

SYM = J

drd0 A’r2igDtiXpAp. (6)

If we take this coupling in five dimensions and an- alyze it from a four-dimensional point of view, we find a Dirac-spinor with both Yang-Mills and Yukawa couplings. By choosing a background field such that A, = 0 for ,u = 0,. . . ,3 we can get a system which has only one Yukawa coupling.

+2iA(x#(x) - e*5@ . &D(x)) 1

. (10)

To introduce a mass term for the fermions, all we have

to do is to add a constant piece to the scalar field. If we shift the scalar field @ by a constant Q -+ Q +m/A we

introduce a mass term into our action. This procedure introduces a term of the form 2ix5m which turns out to be an inconvenience when we want to construct the perturbation expansion (Eq. ( 15) ) . To eliminate this term, we shift x5 + x5 - iem and get

Sy=i SC

dr c+T+em2++.+b+t+b5$5 4 e

+/Y (

fi+ - (x5 - iem)+ >

+zA((ixs+em)@(x) -ie$5$.2@(x)) >

. (11)

202 M. Mondrag6n et al. / Physics Letters 3 351 (1995) 200-205

From this we can integrate out the auxiliary field .“cg, i.e., we eliminate it using its equation of motion: 5

1 sv = -

2 s( dr ;++$+$& +em2+2eAmQ,

+ eA2Q2 - 2iAe*& - SD(x)

t-x (

ai@ + iem& + ieh$# >I

. (12)

If we set @ = 0, we recover the action for the massive,

spinning particle in component notation [ 15,131 or, from Eq. (8)) the action in superfield language [ 1 S] .

So far, we discussed the scalar coupling iA@&,b. It is also possible to find a world-line action which re- produces the one-loop effective action for the Yukawa model with the pseudo-scalar coupling A’Q’$ys@. The only change necessary is the introduction of an- other fermionic superfield X’ = xg + B&&j for the new interaction term. The presence of the two fields 8

and X’ ensures that the terms generated from the scalar and pseudo-scalar interactions do not mix - something that the ys-matrix ensures in the standard field theory treatment.

For a massive, scalar field with a pseudo-scalar cou- pling the resulting action is then

Sps = ; s

drd0A”2(D,X. DoX+ A-‘WDex

+ A-‘X’DeX’ + h’~“irnR + h’i2ih’X’@‘(X)).

(13)

In components (in the x = 0 gauge), after the elimi- nation of the auxiliary fields, this reads

+ e A’2@‘2 - 2iA’e&$ . a@‘(x) >

. (14)

We could also formulate this action analogous to Eq. ( 11) or, from Eqs. ( 12) and ( 14), write down the action for a spinning particle which has both scalar and pseudo-scalar Yukawa couplings.

Now that we have a world-line action for a spinning particle with a Yukawa coupling we want to apply

5 This action differs from the action proposed in [4]. Starting from this suggestion, we were not able to perform the calculations analogous to our one-loop examples presented here.

it to some one-loop calculations. The starting point is always the world-line expression for the one-loop effective action [ 451

mdT r(a) = -2 T DxVx5V*D~5 evTsY. J J (15)

0

This can be used both for deriving rules for the calcu- lation of one-loop n-point functions and as an expres- sion from which one can derive approximations to the one-loop effective action itself.

For any calculation we need rules how to evalu- ate the path integral in Eq. ( 15). In principle, the path integral also includes the integration over the fields e and x. Since infinitesimal changes in those fields are associated with infinitesimal reparametrisa- tions we have a gauge invariance in our system. After treating this with standard methods [ 191 we are left with the conventional integral over T, where T labels

inequivalent circles. This gauge-fixing is the origin of the factor dT/T in Eq. ( 15). The free 2)x path-integral is normalised to (4rT) - d/2, the other free path inte- grals are 1 [ 19,6]. A common [ 13,451, convenient gauge choice is e = 2 and x = 0.

Before proceeding, we separate the centre of mass xc from the embedding coordinate x as

T

x(r) = xc + y(r), with J dry(r) = 0. (16)

0

The remaining path integral can be evaluated using standard methods. From the free part we obtain cor- relation functions for the quantum fields. And the ex- ponential containing the interaction gets expanded ac- cording to the chosen approximation. The interaction part is evaluated using Wick contractions with the cor- relation functions [ 4,5]

(yP”(~1)yy(72)) = -g/““G3(71,72),

G(71972) = 1~1 - ~21 - (71 - 7212

T ’

1 (V(71W’(72)) = -P’G(71,72),

2

Gd71,7-2) = sign(v - 72).

1 (#5(71)@55(72)) = -G ( 2 F 71,721, t 17)

M. Mondraghn et al. /Physics Letters B 351 (1995) 200-205 203

Fig. 1. The two-point function I$’ for the scalar field.

and

(x5(71)x5(72)) =28(71,72)- (18)

The Green’s functions are the ones on the circle for bosonic fields with periodic boundary conditions, and for fermions with anti-periodic boundary conditions. The extra term in the GB results from the background charge required due to the compact nature of the circle. To find the Green’s function ( 18) for x5, one just has to invert the identity operator.

As a simple example indicating how to reproduce the results of a standard Feynman diagram calculation, let us look at the two-point function for the scalar field in the Yukawa model (Fig. 1).

From the standard expression

(2&Vpr +P2)r(2k71 P2) a ’

a2 = ~WPl)~@(PZ)

r(Q) @=O

(19)

we get the world-line expression for the spinor loop correction to the scalar two-point function. From the

Fourier representation

J ddp _. Q,(x) = me zp’“@t(P) (20)

the functional differentiation in Eq. ( 19) automati- cally generates the vertex operator @ + exp( -ip - x) for the scalar field. The expression for the scalar prop- agator correction is then

CWdm +P2)~3mP2)

O”dT = _2 [ F(4,rT) -d/2e-Tnz2 / ddxo ,-ixo.(Pl+Pz)

J ‘ J 0

X d,A2(e-ipl.Y(~)e-ip*.y(7))

0

T T

+ J J dn d72{4~2m2(e--iPI.Y(~1)e-~P2.Yt~2))

0 0

-4A2(k(71)11rS(72))(~~L(71)~“(72))

(21)

Besides the correlation functions from Eq. ( 17) we use the contraction of exponentials

to evaluate (21). The xa-integral produces the momentum conserving

S-function which we use to set p E p1 = -PT. To eval- uate this expression further we use the standard rescal-

ing ri = TUi [ 4,5,20]. This way, all the T-dependence is displayed explicitly and the T-integration can be done, leading to

rC2)(p -p) = -2A2(4+-2 Q, ’ -2r(e-- l)@~~)~-~

+r(E) ,4m2 +p2] jdi (m2 +p2X(1 - x)j-‘)

0

(23)

where we introduce E = 2 - d/2. The result of a standard Feynman parameter calcu-

lation is

rC2)(p -p) = -2A2(4z-)‘-2 @ ’ 1

x J dx( [(2+E)r(+ 1) +r(e)]

0

x (m2+p2x(1 -X))l-•e) (24)

from which we can recover result (23) by judicious integration by parts.

It is worthwhile to note here that we do not have the straightforward connection between the r-intervals of the world-line formalism and the Feynman (Y- parameters. Alternatively, starting from a second- order expression for the fermion one-loop effective action [ 31, it is possible to reorganise the field the- ory perturbation series in a manner analogous to the

204 M. Mondragbn et al. /Physics Letters B 351 (1995) 200-205

cp 6

Fig. 2. The mixed four-point function.

world-line formalism6 (alas, only after performing the integration over the loop momentum).

The difference in the organisation of the perturba- tion series will be even more noticeable in the next example where it is not possible to isolate the con- tribution of a single (first-order formalism) Feynman diagram in the expression generated by the world-line

formalism. To illustrate the point we just made, let us look at

the four-point function (Fig. 2) where, in a standard field theory calculation, six Feynman diagrams con- tribute. The world-line formalism generates only one expression which cannot be divided by restricting the region of integration of the r’s into the contribution corresponding to individual Feynman diagrams. From world-line action ( 14) we immediately get the expres- sion

rC4) (p1 p2 p3 ,p4) = -2 oOdT 3 9 J T ( 4r73 -d/2,-Tm2

x [4kAt2 jdit i dr:

0 0

x (e--ipl.Yle-iP3.Yle-iP2.Y2e-iP4.~)

- SA2Y2jdr, jdRjdrs

0 0 0

x {Cm2 - (~5,1~5,25,2)(~~,1~~,2)P~P;:)

x (e- ip1.yr~-ip3-Y2~-ip2.Y3~-iP4.Y3 >

- @6,1+6,2) (~~,l~v,2) &P:

x (e- ip~.n,-iP3.Y3,-ipz.Y~,-ip4.Y2 H

6 We would lie to thank D. C. Dunbar for discussions on that

point.

+ 16A2At2 j.drt id-jdr3~dr~

0 0 0 0

(25)

Here we use the short-hand notation yi G Y(Q) , and we omitted the momentum conserving &function. The scalar fields are attached to the momenta p1 and p3

while the pseudo-scalar fields are on p2 and ~4. With the contractions ( 17), it is easy to evaluate

this expression. The T-integrations lead again to the I- functions of dimensional regularisation, and the Gn’s produce polynomials similar to polynomials in Feyn- man parameters in the standard field theory calcula- tion. Whereas the T-integration is simple, the remain- ing r-integrations are more complicated - they are of a form similar to scalar bubble, triangle and box inte- grals. To illustrate our point it suffices to extract the 1 /e-term from Eq. (25). The only divergence sits in the first term which is

8A2Af2 1

--IYE) du(m2+(pl.pz+pl.p4+Ps.P2 (4V) 2--E J

0

+p3 .p4)u(l -u))-‘. (26)

This diverges for E + 0 as -A2A12/ ( 2?r2e) _ The same result is easily obtained from the usual evaluation of Feynman diagrams in the first-order formalism. In that case, however, the contributions of different diagrams enter with different signs. Such cancellations do not occur in the world-line formalism - we immediately get the answer for the sum of several Feynman dia- grams. The point we made before should be clear after this example: In general, in the world-line formalism we compute the sum of a class of Feynman diagrams. It is not always possible to identify the contribution of a single Feynman diagram by restricting the integra- tion over the world-line parameters ri*

From the effective action ( 15) we can also generate very elegantly the asymptotic heat-kernel expansion of the effective action [ 21-241.

M. Mondrag6n et al. /Physics Letters B 351 (1995) 200-205 20.5

Instead of working in momentum space, as we did in the previous section, we work now in coordinate space. The field @ and its derivative are now expressed through their Taylor expansion as [ 51

@(x(r)) = eY(+%(.Xa) (27)

about the centre of mass x0. With this we can proceed as before and expand the interaction part of the expo- nential in ( 15)) perform the contractions, and rescale the integration variables. However, instead of perform- ing the T-integration we expand the exponentials of the form exp ( -TGB ( ui, Uj ) di. Jj ) , generated by (22). After arranging the series by powers of T and perform- ing the ui integrals one arrives at a variant [5,25] of the asymptotic heat-kernel expansion, which is partic- ularly well organised [ 261. In a forthcoming publica- tion [27] we will discuss in detail this expansion in the case of the Yukawa model. The calculation of the expansion of the effective action with different meth- ods gives us another check that Eq. (15) indeed is equivalent to the usual effective action in field theory. We verified this for the scalar and pseudo-scalar cou- plings separately as well as for the mixed case.

We managed to construct the world-line action for the Yukawa model both for scalar and pseudo-scalar couplings. For some simple examples we showed how we can reproduce the result of a standard Feynman diagram calculation using our action in the world-line formalism. We performed more calculations to verify the agreement in other cases which, for the sake of brevity, we do not present in this letter.

It has been noted in [6] and further exemplified in [ 281 that the world-line path integral formalism, if used with a global parametrisation, naturally offers the possibility of combining diagrams of different topol- ogy into a single master expression. (For a recent ap- plication of this idea, see also [29] .) It will be inter- esting to see whether this leads to simplifications in the case of the two-loop box diagrams in the Yukawa model, which can be treated with the methods devel- oped in this paper.

We would like to acknowledge the help of D. Flieg- ner and discussions with F? Haberl.

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