renormalisation of the fayet-iliopoulos d-term

27 January 2000

Ž .Physics Letters B 473 2000 102–108

Renormalisation of the Fayet-Iliopoulos D-term

I. Jack, D.R.T. JonesDept. of Mathematical Sciences, UniÕersity of LiÕerpool, LiÕerpool L69 3BX, UK

Received 8 December 1999; accepted 21 December 1999Editor: P.V. Landshoff

Abstract

We consider the renormalisation of the Fayet-Iliopoulos D-term in a softly-broken Abelian supersymmetric theory. WeŽ .show that there exists at least through three loops a renormalisation group invariant trajectory for the coefficient of the

D-term, corresponding to the conformal anomaly solution for the soft masses and couplings. q 2000 Published by ElsevierScience B.V. All rights reserved.

1. Introduction

In Abelian gauge theories with Ns1 supersym-metry there exists a possible invariant that is notallowed in the non-Abelian case: the Fayet-Iliopou-los D-term,

4Lsj V x ,u ,u d usj D x . 1.1Ž . Ž . Ž .HIn this paper we discuss the renormalisation of j inthe presence of the standard soft supersymmetry-breaking terms

j 1 12 i i jk i jL s m f f q h f f f q b f fŽ . ŽiSB j i j k i j6 2

1q Mllqh.c. . 1.2Ž ..2

Let us begin by reviewing the position when thereis no supersymmetry-breaking, i.e. for L s0. ManySB

w xyears ago, Fischler et al. 1 proved an importantresult concerning the renormalisation of the D-termŽ w x. 4see also Ref. 2 . Since it is a H d u-type term, onemay expect that the D-term will undergo renormali-

sation in general. Moreover, by simple power-count-ing it is easy to show that the said renormalisation isin general quadratically divergent. Evidently this

Ž .poses a naturalness problem since if present itwould introduce the cut-off mass scale into the scalarpotential. At the one loop level it is easy to show that

Žthe simple condition Tr YYs0 where YY is the U1

hypercharge and the trace is taken over the chiral.supermultiplets removes the divergence. Remark-

ably, although one may of course easily draw indi-Ž .vidual diagrams proportional for example to

Tr YY 5, YY 7, . . . etc., this condition suffices to allorders.

In the presence of supersymmetry breaking, how-ever, it is clear that j will suffer logarithmic diver-gences. If calculations are done in the componentformalism with D eliminated by means of its equa-tion of motion, then these divergences are manifestedvia contributions to the b-function for m2. It is inthis manner that the results for the soft b-functions

w xwere given in, for example, Ref. 3 . Here we preferto consider the renormalisation of j separately; anadvantage of this is that it means that the exact

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 01484-7

( )I. Jack, D.R.T. JonesrPhysics Letters B 473 2000 102–108 103

results for the soft b-functions presented in Refs.w x Ž w x.4–6 see also Ref. 7 apply without change to theAbelian case. The result for b is as follows:j

bg ˆb s jqb , 1.3Ž .j jg

ˆ Žwhere b is determined by V-tadpole or in compo-j

.nents D-tadpole graphs, and is independent of j . Inˆthe supersymmetric case, we have b s0, where-j

Ž .upon Eq. 1.3 is equivalent to the statement that theŽ .D-term, Eq. 1.1 , is unrenormalised. In the presence

ˆ 2Ž .of Eq. 1.2 , however, b depends on m , h and Mj

Ž .it is easy to see that it cannot depend on b. It isinteresting that the dependence on h and M arisesfirst at the three loop level.

Although in this paper we restrict ourselves to theAbelian case, it is evident that a D-term can occur

Ž .with a direct product gauge group G mG PPP if1 2

there is an Abelian factor: as is the case for theMSSM. In the MSSM context one may treat j as a

w xfree parameter at the weak scale 8 , in which caseˆthere is no need to know b . However, if we know jj

ˆat gauge unification, then we need b to predict j atj

low energies. Now in the D-uneliminated case it ispossible to express all the b-functions associatedwith the soft supersymmetry-breaking terms given in

Ž .Eq. 1.2 in terms of the gauge b-function b , theg

chiral supermultiplet anomalous dimension g and acertain function X which appears only in b 2 ; more-m

Žover in a special renormalisation scheme the NSVZ.scheme , b can also be expressed in terms of g . Itg

is clearly of interest to ask whether an analogousexact expression exists for b . Moreover, there ex-j

ists an exact solution to the soft RG equations form2, M and h corresponding to the case when all thesupersymmetry-breaking arises from the conformal

w xanomaly 9 and it is also interesting to ask whetherthis solution can be extended to the non-zero j case.

The key to the derivation of the exact results forthe soft b-functions is the spurion formalism. Theobstacle to deriving an analogous result for b is thej

Žfact that individual superspace diagrams are as al-.ready mentioned quadratically divergent. This means

that if, for example, we represent a hi jk vertex insuperspace by hi jku 2, then we cannot simply factorthe u 2 out, because it can be ‘‘hit’’ by a superspaceD-derivative. The simple relationship between a

graph with a hi jk and the corresponding one with asupersymmetric Yukawa vertex which holds for thesoft breaking b-functions is thereby lost. We aretherefore unable to construct an exact formula forb ; we do, however, present a solution for j relatedj

to the conformal anomaly solution, but which mustbe constructed order by order in perturbation theory.

2. The b-function for j

Ž .In this section we derive Eq. 1.3 , and show howˆ 2the contributions to b proportional to m can bej

related in a simple way to b .g

We take an Abelian Ns1 supersymmetric gaugetheory with superpotential

1 1i jk i jW F s Y F F F q m F F , 2.1Ž . Ž .i j k i j6 2

and at one loop we have

2 Ž1. 3 3 w 2 x16p b sg Qsg Tr YY , 2.2aŽ .g

i12 Ž1. i i i k l 2 216p g sP s Y Y y2 g YY .Ž . jj j jk l2

2.2bŽ .

Let us consider the D-term renormalisation. Wedefine renormalised and bare quantities in the usualway:

j V d4usmye r2j Z Z1r2 V d4u , 2.3Ž .H HB B j V

where the m factor establishes the canonical dimen-sion for j in ds4ye dimensions. Then writing

an1r2j Z Z sjq 2.4Ž .Ýj V ne

and using

j smye r2j Z , g smer2 gZy1r2 , 2.5Ž .B j B V

it is straightforward to show, using

E g em sy gqb , 2.6Ž .gEm 2

that

Ej em s jqb , 2.7Ž .jEm 2

( )I. Jack, D.R.T. JonesrPhysics Letters B 473 2000 102–108104

where

bg ˆb s jqb , 2.8Ž .j jg

and where

ˆ Lb s La . 2.9Ž .Ýj 1L

Here aL is the contribution to a from diagrams with1 1

L loops. On dimensional grounds,

ˆ 2 ) ) )b sm A g ,Y ,Y qhh A g ,Y ,YŽ . Ž .j 1 2

qMM )A g ,Y ,Y )Ž .3

q Mh) qM ) h A g ,Y ,Y ) , 2.10Ž . Ž . Ž .4

Ž .where we have suppressed i, j, . . . indices for sim-plicity.

By considering the relationship between the origi-nal theory and the one obtained by elimination of theD-field, we can prove a remarkably simple result forA above. The relevant part of the supersymmetric1

Lagrangian is as follows:1 2 ) ) 2Ls D qj DqgDf YYfyf m fq . . . .2

2.11Ž .After eliminating D this becomes

21) 2 2 )Lsyf m fy g f YYf , 2.12Ž . Ž .2

where2 2m sm qgj YY . 2.13Ž .

Ž Ž ..RG invariance of this result gives using Eq. 2.82 2

2 2b m , . . . sb m , . . . q2b j YYŽ . Ž .m m g

ˆ 2qg YYb m , . . . . 2.14Ž . Ž .j

Now b 2 is calculated in the uneliminated La-m

grangian and hence does not contain the ‘‘D-tadpole’’contributions. It is, in fact, given precisely by thepreviously derived formula:

Ei 2 i2b m , . . . s DqX g . 2.15Ž . Ž . Ž .jm jE g

where

E) ) 2Ds2 OOOO q2 MM g 2E g

E Elm n˜ ˜qY qY , 2.16Ž .lm n lm nE Y E Ylm n

E E2 lm nOOs Mg yh , 2.17Ž .2 lm nž /E g E Y

i j ki jk 2 l jk 2 i lk 2 i jlY s m Y q m Y q m YŽ . Ž . Ž .l l l

2.18Ž .Ž .and in the NSVZ scheme2 NSVZ 3 w 2 2 x16p X sy2 g Tr m YY . 2.19Ž .

2Now b is given bym

2 2ˆ2 2b sb m , . . . qg YYb m , . . . . 2.20Ž . Ž . Ž .m m j

In other words, if we calculate in the D-eliminatedformalism, we obtain both ‘‘normal’’ contributionsŽthe ones that would be single-particle irreducible in

.the D-uneliminated case and the D-tadpole contri-butions, which appear in b in the D-uneliminatedj

case.The key now is the result that

2 22 2b m , . . . sb m , . . . . 2.21Ž . Ž . Ž .m m

2This follows simply by substituting for m from Eq.Ž .2.13 and then using the facts that

i j kl jk i lk i jlYY Y q YY Y q YY Y s0 2.22Ž . Ž . Ž . Ž .l l l

by gauge invariance, and

Tr YY 3 s0 2.23Ž . Ž .for anomaly cancellation 1. We then find immedi-ately that:

bg2 2ˆ ˆb m , . . . s2 jqb m , . . . , 2.24Ž . Ž . Ž .j jg

whence

bgTr YYA s2 . 2.25Ž . Ž .1 2g

So if we take the D-tadpole contributions to b , thenj

the terms proportional to m2 will reduce to 2b rg ifg

1 A remark on scheme dependence. The result for X, Eq.Ž .2.19 , applies in the NSVZ scheme, which is one of a class of

Ž .schemes which include the standard perturbative method, DRED ,related by redefinitions of g and M, the ramifications of which

w xare described in Refs. 6 . Now X transforms non-trivially underŽ .these redefinitions, but it can be shown using Eqs. 2.22 and

2 2Ž .2.23 that X is unchanged by the replacement m ™m in anyŽ .member of this class of schemes; consequently Eq. 2.21 always

applies.


we replace m2 by g YY. This result is, in fact, clearfrom a diagrammatic point of view, since the afore-said replacement converts the diagrams into D self-energy graphs, and hence indeed gives rise to b .g

3. The three loop results

Through two loops we have that

Ž .2 2 2 1ˆ w x16p b s2 gTr YY m y4 gTr YY m g q . . .j

3.1Ž .

so we see that in fact only A is non-zero through1

this order. Moreover, since

Ž .2 3 2 3 2 1w x16p b sg Tr YY y2 g Tr YY g q . . .g

3.2Ž .

Ž .we see that Eq. 3.1 is indeed consistent with Eq.Ž .2.25 .

We have calculated several distinct gauge invari-ˆ Ž3.DREDant contributions to b , namely the sets ofj

Ž 4 2 . Ž 2 2 . Ž 3 2 2 .terms that are O gY m , O gY h , O g Y mŽ 3 2 .and O g h . We find that:

ˆ Ž3.DREDb3 j216pŽ .g

mi2 jk l 2s7 Y Y Y m YYŽ . Ž .j li k m

mi2 jk l 2 nq4 Y Y Y m YYŽ . Ž .j kim n l

3 2 2 2w xy Tr Y Y m YY2

5 i k l pm n py Y Y h h YYim n jk l j2

w 2 2 xy2Tr Y h YY

2 w 2 2 3 xq 10y24z 3 g Tr Y m YYŽ .Ž .m n2 i k l 2 2y12 g Y Y m YY YYŽ . Ž .k lim n

q 8y24z 3 g 2Ž .Ž .

=m nik l 2 32Y Y m YYŽ . Ž .k lim n

ji k l 3qh h YY q . . . , 3.3Ž . Ž .ijk l

Ž 2 . i i k l Ž 2 . i i k lwhere Y sY Y , h sh h . Then re-j jk l j jk l

placing m2 by g YY , we obtain

32 Ž3.g 16p Tr YYA s 6 X q12 X q2 X q . . . ,Ž . Ž .Ž .1 1 3 4

3.4Ž .

where

p2 k lm n 2 4 4w xX sg Y P YY Y , X sg Tr P YY ,Ž . m1 l k n p 3

2 w 2 2 xX sg Tr P YY , 3.5Ž .4

in precise agreement with the result for b Ž3., given ingw x10 , which for an Abelian theory is

32 Ž3.DRED16p b sg 3 X q6 X qXŽ . �g 1 3 4

6 w 4 xy6 g QTr YY . 3.6Ž .4Ž 5 2 .We have not calculated the O g m contributions

ˆ Ž3. 7Ž .to b , which will produce the O g terms in Eq.j

Ž .3.6 .

4. The conformal anomaly trajectory

The following set of equations provide an exactsolution to the renormalisation group equations forM,h,b and m2:

bgMsM , 4.1aŽ .0 g

hi jk syM b i jk , 4.1bŽ .0 Y

bi j syM b i j , 4.1cŽ .0 m

dg ii j212 < <m s M m . 4.1dŽ . Ž .j 02 dm

Moreover, these solutions indeed hold if the onlysource of supersymmetry breaking is the conformalanomaly, when M is in fact the gravitino mass.0

This set of soft breakings has caused considerableinterest; but there are clear difficulties for the MSSM,since it is easy to see that sleptons are predicted to

Ž .2have negative mass . Most studies of this scenariohave resolved this dilemma by adding a constant m2 ,0

presuming another source of supersymmetry break-


ing. A non-zero j alone is not an alternative, unfor-Ž .tunately, as is easily seen from Eq. 2.13 ; the two

selectrons, for example, have oppositely-signed hy-percharge so one of them at least remains with

Ž .2negative mass .Ž .It is immediately obvious that, given Eq. 4.1 ,

there is a RG invariant solution for j through twoˆŽ .loops for b given by:j

22 2< <16p jsg M Tr YY gyg , 4.2Ž .Ž .0

since differentiating with respect to m and using Eq.Ž . Ž . Ž .4.1d leads at once to Eqs. 2.8 and 3.1 . Interest-ingly, however, this result for j Õanishes at leadingand next-to-leading order, since one easily demon-strates that

Ž .1Tr YYg s0 4.3Ž .and

2Ž . Ž .2 1Tr YYg sTr YY g , 4.4Ž .Ž .using the result for g Ž2., which is

22 Ž2. i16p gŽ . j

pm p i 2 2 i ns yY Y y2 g YY d PŽ . jjm n n p

i4 2q2 g YY Q. 4.5Ž . Ž .j

It is interesting to ask whether the trajectory can beextended beyond two loops, and whether it in factcontinues to vanish order by order. We have shown

Ž .that there is indeed a generalisation of Eq. 4.2 to atˆŽ .least three loops for b , and that at this order thej

result for j is non-zero.Our result is as follows:

16p 2j DRED

2< <g M0

2sTr YY gygŽ .y32 2y24z 3 g 16pŽ . Ž .

=i3 2 3 jk l mw xTr YY P q YY Y Y PŽ . j i k m l

y3 i2 jk l m nq2 16p YY Y Y P PŽ . Ž . j im n k l

3 2 3 2w x w xy2Tr YYP y4 g Tr YY P q . . . .

4.6Ž .

ˆNote that our partial b expression determines onlyj

Ž 6. Ž 4 2 .the O Y and Y g terms on the RHS of Eq.Ž .4.6 ; it is interesting that they can be written interms of invariants involving P. After this is donewe have chosen to ‘‘promote’’ remaining Y 2 factorsto 2 P; the difference thereby introduced dependsonly on terms we have not calculated. It is easy to

Ž 8 6 2 .verify that for Y and Y g terms the result ofEtaking m of this expression is identical to thatEm

Ž . Ž .obtained by substituting Eqs. 4.1b and 4.1d inŽ . Ž .Eqs. 3.1 and 3.3 . This is a non-trivial result in

that the number of candidate terms for inclusion inŽ .Eq. 4.6 is considerably less than the number of

Ž . Ž .distinct terms which arise when Eqs. 3.1 and 3.3are placed on the RG trajectory. We therefore con-

ˆ Ž3.jecture that the trajectory holds for the full bj

calculation, and extends to all orders.2 Ž3. w xUsing the result for g from Ref. 11 , we find

that32 216p Tr YY gygŽ . Ž .

s I q12z 3 I qO Y 8 ,Y 6 g 2 , . . . , g 8 , 4.7Ž . Ž .Ž .1 2

wherei13 jk l m nw xI sTr YYP y YY Y Y P PŽ . j1 im n k l2

2 w 3 2 x 4 w 3 xq2 g Tr YY P y2 g QTr YY P ,i2 3 jk l mI sg YY Y Y PŽ . j2 i k m l

2 w 3 2 x 4 w 5 xqg Tr YY P q2 g Tr YY P . 4.8Ž .Ž . Ž .Comparing Eq. 4.6 with Eq. 4.8 leads us to the

conjecture that on the RG trajectory j DRED is givenat leading order by the expression

j DREDy42s 16p y3I y12z 3 IŽ . Ž .Ž 1 22< <g M0

6 w 5 x 4 w 3 xqn Qg Tr YY qn Qg Tr P YY .1 2

4.9Ž .

where n ,n are undetermined. Note that other in-1 2

variants which might in principle have appeared inŽ .Eq. 4.9 are in fact ruled out from consistency with

2 Ž3. DRED Ž3. NSVZNotice that Tr YYg sTr YYg in the Abelianw x w xŽ .case, by virtue of Eq. 2.23 .


ˆ Ž3.DREDb ; and in fact our conjecture is equivalent toj

ˆ Ž3.DREDthe following form for b :j

ˆ Ž3.DREDb3 j216pŽ .g

2 Ž .2 2 2sy6 16p Tr YY m gŽ .5 )w x w xy4Tr WP YY y Tr HH YY2

w 2 2 x 2 w 3 xq2Tr P m YY y24 g z 3 Tr W YYŽ .1 2 ) 3w xq 12z 3 y n g Tr M HYY qc.c.Ž .Ž .22

4 ) w 5 xq6 y48z 3 qn g MM Tr YYŽ .Ž .1

2 ) w 3 xq4n g MM Tr P YY , 4.10Ž .2

w xwhere 3

i1 1i 2 2 2 2 2W s Y m q m Y qhŽ . jj 2 2

r ii p q 2 2 ) 2q2Y Y m y8 g MM YYŽ . Ž .q jj p r

4.11Ž .

and

ii i k l 2 2H sh Y q4 g M YY . 4.12Ž . Ž .jj jk l

It is interesting to note that the particular valueˆ Ž3.DRED iŽ .n s48z 3 makes b vanish if P s0,1 j j

12 i ) i i jk i jkŽ .m s MM d , and h syMY . These rela-j j3

Ž .tions arise at leading order in finite softly-brokenŽtheories of course in the Abelian case considered

.here we cannot achieve a finite theory as Q/0 .ˆ Ž3.It is natural to ask what the result for b is inj

Žthe NSVZ scheme, which is obtained at the relevant. w xorder by the redefinitions 4

2 12 3 2w x16p d gsy g Tr P YY ,Ž . 2

2216p dMŽ .22 2 2 2w xsyMg Tr P YY y2 g Tr YYŽ .½ 5

j1 2 i k l 2q g h Y YY . 4.13Ž . Ž .ijk l2

It is straightforward to show that in order to obtainŽ . Ž .the results Eqs. 2.8 and 2.25 in the NSVZ scheme,

we must also redefine j as follows:

2 12 2 2 2w x w x16p djsy g Tr P YY jygTr m P YY .Ž . 2

4.14Ž .

Ž .The effect of this is to replace Eq. 4.9 by

j NSVZy42s 16p y4I y12z 3 IŽ . Ž .Ž 1 22< <g M0

6 w 5 x 4 w 3 xqn Qg Tr YY qn Qg Tr P YY ,.1 2

4.15Ž .Ž .and Eq. 4.10 by

ˆ Ž3.NSVZb3 j216pŽ .g

2 Ž .2 2 2sy4 16p Tr YY m gŽ .5 )w x w xy 2Tr WP YY qTr HH YYŽ .2

2 w 3 xy24 g z 3 Tr W YYŽ .1 2 ) 3w xq 12z 3 y n g Tr M HYY qc.c.Ž .Ž .22

4 ) w 5 xq6 y48z 3 qn g MM Tr YYŽ .Ž .1

2 ) w 3 xq4n g MM Tr P YY . 4.16Ž .2

It is interesting to note that with the particular valuesŽ . Ž . Ž . Ž .n s48z 3 , n sy24z 3 , Eqs. 3.1 and 4.161 2

are consistent with the following expression

ˆ NSVZbj2 2 2w x16p s2Tr YY m y4Tr YY m gg

E5 2q DqX y Tr g YYž 2ž /E g

y12 2 3q24z 3 g 16p Tr g YY ,Ž . Ž . /4.17Ž .

Ž . Ž .where D and X are defined in Eqs. 2.15 , 2.16 ,Ž . Ž . Ž .2.17 , 2.18 and 2.19 .

Ž .We hope to test our conjectures Eqs. 4.9 andˆ Ž3.Ž .4.10 by completing the calculation of b ; we alsoj

plan to discuss scheme dependence in more detail,and extend our results to a gauge group with directproduct structure and one or more abelian factorsŽ .such as the MSSM .

Note added

Since this paper was submitted we have calcu-ˆ Ž3.DREDlated the contributions to b of the formj

3 w ) 3 x 3 ) w 3 xg Tr M HYY qc.c. and g MM Tr P YY .These are both consistent with the result n s0.2


Ž .Although this means that the z 3 terms cannot, inŽ .fact, be written in the form suggested in Eq. 4.17 , it

does provide strong evidence in favour of our resultŽ .Eq. 4.10 . We feel that our demonstration that j

has a RG trajectory related to the conformal anomalyone is intriguing, and worthy of further investigation.

Acknowledgements

This work was supported in part by a ResearchFellowship from the Leverhulme Trust. We thankJohn Gracey for conversations.

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renormalisation of the fayet-iliopoulos d-term

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