1099

The perturbative β-function in theZumino model

R. Clarkson and D.G.C. McKeon

Abstract: We consider the perturbativeβ-function in a supersymmetric model in four-dimensional Euclidean space formulated by Zumino. It turns out to be equal to theβ-functionfor N = 2 supersymmetric Yang–Mills theory despite differences that exist in the two models.

PACS No.: 12.60Jv

Résumé: Nous étudions la fonctionβ perturbative dans un modèle supersymétrique sur unsupport 4-D euclidien tel que proposé par Zumino. Nous la trouvons identique à la fonctionβ de la théorie supersymétrique de Yang–Mills avecN = 2 et ce, malgré les différences entreles deux théories.

[Traduit par la Rédaction]

1. Introduction

Supersymmetry in Euclidean space is quite distinct from supersymmetry in Minkowski space [1].There is in fact no direct analogue in Euclidean space of theN = 1 supersymmetry algebra in Minkowskispace, as Majorana spinors cannot be defined in Euclidean space. The simplest supersymmetry algebrain Euclidean space bears some resemblance to theN = 2 supersymmetry algebra in Minkowski space;they do differ though, in particular, the central charge in Minkowski space forms a lower bound to theenergy while in Euclidean space it forms an upper bound to the energy.

N = 2 supersymmetric Yang–Mills theory in Minkowski space (SSYM2) is invariant under thesymmetries present in theN = 2 supersymmetry algebra [2], while the Euclidean space supersymmetricmodel of Zumino (Z) [3] possesses the corresponding invariances in Euclidean space [1]. On shell, bothmodels contain two scalars, a vector, and four complex fermionic degrees of freedom (a pair of Majoranaspinors in SSYM2 and a Dirac spinor inZ). However, there are, in fact, real distinctions between themodels as well; inZ, the kinetic terms for the scalars have different signs while they are the same inSSYM2; the quartic scalar couplings have different signs in the two models and theYukawa pseudoscalarcoupling differs by a factor ofi. TheZ-model in Euclidean space was not formulated to reproduce ananalytic continuation of Green’s functions of the SSYM2 model in Euclidean space; indeed it was shownin ref. 4 that analytic continuation of theWess–Zumino model from Minkowski space to Euclidean spaceyields a model in which Hermiticity is somewhat unconventional. We also note that when one considers

Received March 28, 2001. Accepted July 15, 2001. Published on the NRC Research Press Web site on Septem-ber 4, 2001.

R. Clarkson and D.G.C. McKeon.1 Department of Applied Mathematics, University of Western Ontario,London, ON N6A 5B7, Canada.

1 Corresponding author Telephone: (519) 679-2111, ext. 88789; FAX: (519) 661-3523;e-mail: DGMCKEO2@.UWO.CA

Can. J. Phys.79: 1099–1104 (2001) DOI: 10.1139/cjp-79-8-1099 © 2001 NRC Canada

1100 Can. J. Phys. Vol. 79, 2001

the Yukawa coupling of scalars to spinors in three dimensions, theβ-function in 2+ 1 dimensions,surprisingly, differs from that in 3+ 0 dimensions [5].

These observations have motivated us to actually test to see by how much the models differ bycomputing theβ-function in theZ model to see if it is the same as the knownβ-function in the SSYM2model [6]. If they were to differ, then the differences between these models would be greater than whatone might expect based on the observation that the SSYM2 model [7] and theZ model [1,8] can bederived by dimensionally reducing theN = 1 supersymmetric Yang–Mills theory in 5+ 1 dimensionsto 3+ 1 and 4+ 0 dimensions, respectively.

In this paper, an explicit calculation is made of the one-loopβ-function in theZ model. Beyondone-loop order, it is demonstrated that the result of computing Green’s functions in Euclidean space withtheZ model yields the sameβ-function as is obtained by computing Green’s functions in Minkowskispace in the SSYM2 model. This is done without explicitly calculating Feynman diagrams.

In performing the calculations, we use dimensional regularization [9] even though, strictly speaking,supersymmetry does not hold inn-dimensions. It has been demonstrated that this does not present adifficulty at one-loop order. (See ref. 10 for a discussion of regularization of supersymmetric theories.)

2. The β-function to one-loop order

The actions we consider are

SM =∫

d4xM

[−1

4FaµνF

aµν + 12

(Dabµ A

b)2 + 1

2

(Dabµ B

b)2 − i

2 ψa

i γ ·Dabψbi

+ig ψai f abc(Ab − iBbγ5

)ψci −1

2 g2(f abcAbBc

)2], ηµν = (+ − −−) (1a)

and

SE=∫

d4xE

[−1

4FaµνF

aµν+ 12

(Dabµ A

b)2− 1

2

(Dabµ B

b)2 − i

2

(ψ+a (γ ·Dabψb

)−(Dabψ+b)·γψa)

+igψ+af abc(Ab − Bbγ5

)ψc +1

2 g2(f abcAbBc

)2], ηµν = (+ + ++) (1b)

for the SSYM2 andZ models respectively. As usual, we have

Faµν = ∂µVaν − ∂νV

aµ + gf abcV bµV

cν (2a)

and

Dabµ = ∂µδab + gf apbV pµ (2b)

Feynman rules follow immediately from (1). All couplings are characterized by a single couplingconstantg; the β-function associated with its renormalization in theZ model will be deduced byconsidering divergences in theA− ψ − ψ vertex.

Working with bare fields and the bare couplinggB , the one-loop correction to the fermion self-energyis given in the Feynman gauge by

∑= −g

2BC2(G)

4π2ε6p + (finite) (3)

whereε = 4 − n andf amnf bmn = C2(G)δab. Similarly, the self-energy of the scalar fieldA is given

in the Feynman gauge by

5 = −g2C2(G)

2π2εp2 + (finite) (4)

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Clarkson and McKeon 1101

and the correction to the vertexA− ψ − ψ is given by(−gf man3) with

3 = −g2C2(G)

4π2ε+ (finite). (5)

Using these results, theβ-function now can be computed using the approach given in ref. 11. Therelations

ψR = √Z2ψB (6a)

AR = √Z3AB (6b)

can be used to render the two-point functions〈ψRψR〉 and〈ARAR〉 finite. Since at tree level,〈ψBψB〉 =+ 1

6p , 〈ABAB〉 = − 1

p2 ,

〈ψRψR〉−1 = Z2(6p +6) (7a)

〈ARAR〉−1 = −Z3(p2 −5) (7b)

From (3) and (7a), we find that

Z2 = 1 + g2BC2(G)

4π2ε(8a)

Z3 = 1 − g2BC2(G)

2π2ε(8b)

For the three-point function we have

〈ARψRψR〉 = Z2Z1/23

[+igBf man − igBf

man

(g2BC2(G)

4π2ε

)](8c)

If now

gB = gRµε/2

(1 + a11g

2R

ε+ . . .

)(9)

Then by (8),〈ARψRψR〉 is finite provided

a11 = C2(G)

4π2

Since (9)gB is independent ofµ, we must have

µ∂gR

∂µ= −a11

2g3R + . . .

= −C2(G)g3R

8π2 (10)

This is identical to the perturbativeβ-function to one-loop order in the SSYM2 model [6].

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1102 Can. J. Phys. Vol. 79, 2001

3. The β-function beyond one-loop order

It is a text book exercise to obtain Feynman rules for the perburbative calculation of Green’s functionsonce one is presented with a classical Lagrangian. It is useful to recall some of their features to relateGreen’s functions computed in Euclidean space for theZ model and Green’s function computed inMinkowski space for the SSYM2 model.

Suppose we have a Lagrangian for a generic fieldφA that is of the form

L = 1

2φA�φA + λ

N !φNA (11)

If we are in Euclidean space, the generating functional is

ZE[J ] =∫DφA exp

∫dx

[1

2φA�φA + λ

N !φNA + JφA

](12)

which yields the propagator

〈φAφA〉E = −�−1 (13a)

and the vertex

〈φNA 〉E = +λ (13b)

However, in Minkowski space, the generating functional becomes

ZM[J ] =∫DφA exp i

∫dx

[1

2φA�φ + λ

N !φNA + JφA

](14)

so that we have the propagator

〈φAφA〉M = i�−1 (15a)

and the vertex

〈φNA 〉M = iλ (15b)

From (13) and (15), we see that an overall factor of

(−i)I (+i)V (16)

distinguishes a Feynman diagram withI internal lines andV vertices in Euclidean space from thecorresponding diagram in Minkowski space.

Furthermore, each loop momentum integral with degree of divergenced generates an additionalcontribution of

i(−1)d (17)

This can be seen from the form of the standard Feynman integral inn-dimensions; in Euclidean spaceit is∫

dnkE

(2π)n(k2)a

(k2 +m2)b= 1

(4π)n/2

(m2)n/2+a−b 0(b − a − n

2)

0(b)

0(n2 + a)

0(a)(18a)

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Clarkson and McKeon 1103

while in Minkowski space it is∫dnkM

(2π)n(k2)a

(k2 −m2)b= i(−1)a−b 1

(4π)n/2

(m2)n/2+a−b 0(b − a − n

2)

0(b)

0(n2 + a)

0(a)(18b)

Consequently, anL-loop diagram will result in an additional factor of

(i)L(−1)D (19)

whereD is the overall degree of divergence of the diagram.Together (16) and (19) show that a Feynman amplitude in Euclidean space and a Feynman amplitude

in Minkowski space differ by a factor of

f = iL−I+V (−1)D (20)

Standard topological arguments show that

L = I − V + 1 (21)

so that (20) reduces to

f = i(−1)D (22)

which is independent ofL, I , andV . For a supersymmetric theory, all divergences are at most log-arithmic, so thatD ≤ 0. Consequently, for the specific case of computing the amplitudes needed todetermine theβ-function in the SSYM2 andZ models discussed above, we can setD = 0. Thus by(22), relevant graphs for the〈Aψψ〉 vertex used to computeβ differ by a factor ofi in the two models;hence the expression forβ is the same in the two models, provided they had the same form for theclassical action.

However, there are in fact differences between the forms of the actions in (1a) and (1b). Thepropagator for the pseudoscalar fieldBa in the two models differs by a factor of(−1), and verticesto whichBa contributes acquire a factor of(−i) for each externalBa leg. Consequently, the overalldifference between analogous diagrams in Euclidean and Minkowski space containingIB propagators

associated with the fieldBa is[(−1)(i)2

]IB = 1; i.e., the fact that the Lagrangians in (1a) and (1b) arenot the same will not alter the conclusion that analogous Green’s functions in the two models will differonly by an overall factor ofi.

The gamma matrix algebra in the models of (1a) and (1b) is{γµ, γν

} = 2ηµν with ηµν being themetrics (+ − −−) and (+ + ++), respectively; consequently6 p2 = p2 in both models. This meansthat theγ matrix algebra for these two models will not affect theβ-function. In ref. 5, a difference wasin fact found between theβ-function for aYukawa model in three-dimensional Euclidian space and theanalogous model in three-dimensional Minkowski space. It was seen that this is a result of the natureof the gamma matrix algebra in the two spaces within the context of the conventions used; altering theconventions will not change the conclusion that theβ-function in the two models is not the same.

4. Discussion

We have demonstrated that the perturbativeβ-function is identical in the Minkowski space SSYM2model and the EuclideanZ model. Explicit calculations have shown this to be true at one-loop order.This persists beyond one-loop order (where it is known that theβ function for the SSYM2 modelvanishes [12]). However, it is not clear whether the twoβ-functions are the same in the nonperturbativeregime (where it has been worked out in the SSYM2 model [13]). It would also be interesting to see iftheβ-function forN = 4 supersymmetricYang–Mills theory (known to be zero) is the same as that fora corresponding Euclidean space model [1]. The former model is obtained by dimensional reduction ofN = 1 supersymmetric Yang–Mills theory from 9+ 1 to 3+ 1 dimensions while the latter comes froma dimensional reduction from 9+ 1 to 4+ 0 dimensions.

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1104 Can. J. Phys. Vol. 79, 2001

Acknowledgements

We thank the Natural Sciences and Engineering Research Council of Canada for financial support.

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