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Feb 1998 IFUP{TH 63/97

hep-ph/9802446

Next-to-leading order corrections

to gauge-mediated gaugino masses

Marco Picariello and Alessandro Strumia

Dipartimento di Fisica, Universita di Pisa and

INFN, sezione di Pisa, I-56126 Pisa, Italia

Abstract

We compute the next-to-leading order corrections to the gaugino masses M in gauge-

i

mediated mo dels for generic values of the messenger masses M and discuss the predic-

tions of uni ed messenger mo dels. If M < 100 TeV there can b e up to 10% corrections

to the leading order relations M / . If the messengers are heavier there are only

i i

few % corrections. We also study the messenger corrections to gauge coupling uni -

cation: as a result of cancellations dictated by sup ersymmetry, the predicted value of

the strong coupling constant is typically only negligibly increased.

1 Intro duction

The \gauge mediation" scenario for sup ersymmetric particle masses [1] can b e realized in reasonable mo dels [2, 3].

Furthermore, with a uni ed sp ectrum of messenger elds, it gives rise to some stable and acceptable prediction

for the sp ectrum of sup ersymmetric particles. One of these predictions is the `uni cation prediction' for the

gaugino masses M (i =1;2;3): at one-lo op order the RGE-invariant ratio   M = is the same for all the

i i i i

N

three factors of the SM gauge group G = G = U(1) SU(2) SU(3) . This prediction is suciently

SM i Y L C

i

stable that it is interesting to compute it with more accuracy.

A detailed computation allows a comparison with the gaugino sp ectrum predicted by the alternative scenario

known as \uni ed supergravity" [4]. Uni cation relations for the  (E ) are infact the more stable prediction

i

of this second scenario: gauge couplings and gaugino masses could receive sizeable GUT threshold corrections;

but these corrections largely cancel out in the ratios  (M ), since and M have the same one-lo op RGE

i GUT i i

dr scheme [5] are (including evolution. The testable predictions for the low-energy running  ratios in the

i

NLO RGE corrections [6], but neglecting p ossible unknown O (%) GUT-scale threshold e ects [7])

 (M )  (M )

2 Z 1 Z

 1:02;  0:97: (1)

 (M )  (M )

2 Z 3 Z

This should be compared with the corresp onding prediction in gauge-mediation mo dels for the ratios  = ,

i j

plotted in g.s 2 and 3.

The computation of gaugino masses with NLO precision is done in sections 2 and 3 for generic values of the

messenger mass M . It requires the following main steps:

1. compute the renormalized running gaugino masses, M (E )atE M, in the e ective theory without

i H H

messengers. 1

2. compute the running gaugino masses, M (E )atE M , evolving M (E )down to the Fermi scale

i L L Z i H

with 2 lo op RGE equations.

p ole

3. use M (E ) to compute directely measurable quantities, like the gaugino p ole masses, M , including

i L

i

all 1 lo op e ects at the electroweak scale.

The computation necessary for step 1 is done in section 2. We will employ sup ersymmetric dimensional regu-

dr scale, . The RGE necessary for step 2 (recalled larization, so that the renormalization scale E will b e the

in app endix B) can b e read from the literature [6]. The one-lo op expressions for p ole gaugino masses in terms

of running parameters are also well known [8]. Since various unmeasured and unpredicted parameters (like the

so-called -term) would enter the nal step 3, we prefer to show our nal predictions for the running MSSM

gaugino masses renormalized at  = M , without including the gauge corrections at the electroweak scale.

L Z

We will compute these predictions in uni ed messenger models. One more step is necessary to imp ose the

uni cation constraints on the messenger sp ectrum, namely

0. compute the messenger sp ectrum, M (E ), evolving the uni ed M (M )down to the messenger scale

n H n GUT

E  M with 2 lo op RGE equations.

H n

The necessary RGE equations are given in app endix B. In sec. 3 we study the predictions of gauge mediation

mo dels with an uni ed messenger sp ectrum. If the messenger sp ectrum is only negligibly splitted by sup ersym-

metry breaking e ects, the NLO corrections to the LO uni cation-like relations M / are around few % and

i i

numerically not much di erent from the ones present in uni ed sup ergravity mo dels. Larger e ects (up to 10%)

<

can b e present if the messengers are very light, M 50 TeV.



We also study the corrections to gauge coupling uni cation due to the presence of messenger elds b elow the

uni cation scale. Messenger threshold e ects largely cancel messenger corrections to twoloopRGE running, as

dictated by sup ersymmetry [9, 10].

2 Computation

In this section we do the computations necessary for step 1. We assume that the messenger elds sit in real

L

representations R = R of the SM gauge group and have a sup ersymmetric mass term M together with

n n

n

a non-sup ersymmetric mass term F . For the moment we assume that the messengers lie in self-conjugate

n



complex representations, R = X  X . See app endix A for a more detailed discussions of the notations, of

n n n

the mo del, of its sp ectrum, and of the relevant Lagrangian.

<

In order to compute the running gaugino masses M ()at M in the e ective theory we rst compute

i n



the gauge-indep endent pole gaugino masses in the twoversions of the theory. The computation in the full theory

(with messengers) is done in section 2.1 | the computation in the e ective theory (with messengers integrated

out) is done in section 2.2. Requiring that the two theories describ e the same gaugino masses up to second

order in we get, in section 2.3 the gaugino mass terms in the e ective theory.

i

dr regularization [5] in b oth versions We employ the Feynman-Wess-Zumino gauge and the sup ersymmetric

dr values of the gauge couplings and of the gaugino masses M can b e converted of the theory. The nal

i i

into the ms ones (i.e. the ones obtained with nave dimensional regularization) using

C (G)

); M ) = (1 = M (1 + C (G)

ms dr ms dr

3 4 4

N

where the group factors are de ned as follows. For each representation R of a gauge group G = G we de ne

i

i

a

the \Dinkin index" T (R ) and the \quadratic Casimir" C (R ) in terms of the generators T as

i i

Ri

X

ab b a a a

= T (R )  : (2) T = C (R )1I; Tr T
T T

i ij i

Rj Ri Ri Ri

a



With generators canonically normalized so that T (n)=T(n)=1=2 for the fundamental n representation of a

SU(n) group, the quadratic Casimir of the adjoint representation of a SU (n) group is C (G)=T(G)=n, while 2

graph and its value V graph and its value V

10 1

~

XX C (X)( ) X X [C (X )  C (G)](2=" +4)

j j j j ij i

3 2

g

4 1

g ~

XX C (X) (1=" +1) XX [C (X)  C (G)](4)

j j j j ij i

3 2

~ ~

X X C (X )(3=" 2=3) X C (G)(4=" 1)

j j i i

g

8

~ ~ ~

X X C (X ) (2=" 1) X C (G)(1=" 1)

j j i i

3

g

5 1 4

0

~ ~

X X C (X )( + )  C (G)(4=" +4)

j j i i ir

3 " 3

(1)

2

Table 1: Contributions M = M V =4 of the single graphs in the limit F  M .

i

i

C (G)=0for G a U(1) factor. The values of the co ecients for the SM gauge group and for representations



contained in 5  5 and 10  10 representation of SU(5) are given in table 2.

2.1 Computation in the full theory

The NLO correction to the p ole gaugino masses are given by the ten two-lo op diagrams shown in gure 1 and

some one-lo op renormalization factor. It is convenient to separate the renormalization factors due to the light

MSSM lo ops from the ones due to messenger lo ops. We write the various contributions to the p ole gaugino

masses as

X X X

F

n i

(1) (1) (1) (1)

full

^ ^ ^

T (R )^g (x )=M + O (") (3) M j = M + M M = ; M 

i n 1 n i

p ole i

i i in i

4 M

n

n n

P

2

where x = F =M and the sum extends over all the messengers. The one-lo op function g^ (x)  g (x)+

n n 1 1

n

n

2

"g (x)+O(" )is[11]

1

 

2 2

 

ln(1 + x)  x 1+x

2 4

ln(1 + x) 1+" 1 1+"ln +(x !x)+O(" )=1+ +O(x ;") (4) g^ (x)=

1

2 2

x 2 M 6

All parameters are unrenormalized (`bare'). Here we list all the contributions to the p ole gaugino masses.

 The contribution given by the sum of the two lo op diagrams of g. 1. They give

(

 

X

F 4 4

i n i

2loop

M = T(R ) C (G) g^ (x )( + g )+2g (x ) +

i n i 1 n IR 2 n

i

4 M 4 " "

n uv ir

n

)

   

3

"

0 2

X

2x g^ (x ) 

j n n

1

+ C (X ) g (x ) 4g (x ) (5)

j n C n 1 n

2

4 " M

uv

n

j=1

0

whereg ^ (x) is the derivativeof^g (x) with resp ect to x, and the functions g and g are

1 2 C

1

 

ln(1 + x)

g (x) = 2(1 + x) + ln (1 x)+(2x+2+1=x) ln(1 + x) + (6 a )

2

2

2x

 

1 2x

+ Li ( )2Li (x) +(x !x)

2 2

x 1+x

 

ln(1 + x)

g (x) = 8+6x + ln(1 x)+(1x+2=x)ln(1 + x) +(x !x) (6 b )

C

2

x

2

In the limit F  M (x ! 0) g (0) = g (0) = 1 and g (0) = 0. We have denoted as 1=" an 1="

1 2 C uv

ultraviolet (UV) p ole, and as 1=" a p ole of infrared (IR) origin. In all the graphs it is p ossible to set the

ir

external gaugino momentum to zero, except in the infrared divergent  graph of g. 1. It is convenient

to split it into a part computed with zero external momentum, that contributes to g , plus the remainder,

2

that gives the g term in eq. (5). The value of g coincides with the `non nave part' of the `asymptotic

IR IR

1

expansion' in the external gaugino momentum, and can b e seen as the contribution of the  graph with

1

The general technique of asymptotic expansions of Feynman diagrams is describ ed in [12]; a much simpler discussion, sucient

for the purp oses of this computation, can b e found in [13], where an accurate distinction b etween UV and IR divergences is made. 3

g

XX XX

g

~ ~ ~ ~

XX XX

g

0

~ ~

XX 

~

X X

g

~ ~

X X X X

Figure 1: The diagrams that contribute to the NLO gauge corrections to gauge mediated gaugino masses. The

thick continuous (dashed) lines represent the fermionic (bosonic) messengers. The thin wavy (continuous) lines

represent the gauge bosons (gauginos).

the heavy messenger lo op contracted to a p oint. This technical detail is useful, b ecause a corresp onding

one-lo op diagram gives the same contribution to the e ective theory, so that we do not need to compute

g .

IR

 On-shell renormalization of the gaugino wave function: the renormalized gaugino eld is  =

i

0 1=2

(1 + z + z )
 j with

i i bare

i

 

2 2 2

X

1  x +(1x )[ln(1 x ) + ln(1 + x )]

n n i

n n

0

T (R ) +ln : + = z

i n

i

2 2

4 " M 2x

n n

n

0

Wehave separated this correction into two parts: z due to all the MSSM particles, and z due to messenger

0

z z

lo ops only. The corresp onding corrections to the gaugino masses, M + M , are obtained expressing

i i

0

(1)

z 0

^

the LO result in terms of the renormalized eld  : M = M
z . We do not need to sp ecify the

i

i i

i

MSSM part b ecause it is the same in b oth versions of the theory. 4

rep. C C C T T T

1 2 3 1 2 3



d  d 2=15 0 8=3 2=5 0 1



L  L 3=10 3=2 0 3=5 1 0



Q  Q 1=30 3=2 8=3 1=5 3 2

u  u 8=15 0 8=3 8=5 0 1

e  e 6=5 0 0 6=5 0 0



Table 2: Values of the group factors for the G fragments of the 5  5 and 10  10 SU(5) representations.

SM

 Renormalization of the gauge couplings: wecho ose to express the bare gauge couplings of the ful l

theory, g j , as function of the quantum-corrected gauge couplings in the e ective theory renormalized

i bare

2

in the dr scheme, g (). De ning  g =4 and including the messenger thresholds e ects we nd

i i

i

 

0 2

X

  ln(1 x )+ln(1+x ) 1

i n n i

0

()= j   ; T (R ) +ln : =

i i bare i i i n

2

4 " M 6

i

n

n

0

Wehave separated the correction to into two parts:  due to all the MSSM particles, and  due to

i

0

 

messenger lo ops only. The corresp onding corrections to the gaugino masses, M + M , are obtained

i i

0

(1)

 0

^

expressing the LO result in terms of the renormalized MSSM gauge coupling (): M = M
 = .

i i i

i

i

We do not need to sp ecify the MSSM part b ecause it is the same in b oth versions of the theory. Notice

0 0

 z

that the messenger contribution M + M is UV nite.

i i

 Renormalization of F and M . We employ dr renormalization: F and M are de ned as their bare

values plus the p ole parts of their quantum corrections. As a consequence of the non-renormalization

theorem, the parameters M and F renormalize in the same way:

n n

X

j

2C (X ) fF ;M g fF ;M g = 1

j n n n bare n n

4

j

The corrections to the gaugino masses are obtained expressing the bare parameters in terms of the renor-

malized ones in the one lo op result:

 

0

X X

F 2x g^ (x )

n j i n n

F;M

1

M = T (R ) C (X ) 4g (x )+ : (7)

i n j n 1 n

i

4 M 4 "

n uv

n

j

2 2

The rst term of (7) derives from ln  =M in (4) and, for x = 0, cancels the NLO correction to M

i

prop ortional to pro duced by the two-lo op diagrams.

j

As an aside remark, it could be of interest to know that the NLO squared p ole mass of the lightest scalar

2

messenger, M ,is

n



2

2

X

M



j

n

)+ = 1 C (X ) 2(x 2)(2 + ln

j n n

2 2

M F 4 M

n

n n

j



2 2

2x 33x 2x

n

n n

+ ln(1 x )+ ln x (1 + x ) ln(1 + x )

n n n n

x 1 x 1

n n

when expressed in terms of dr parameters.

2.2 Computation in the e ective theory

In the e ective theory wehave to compute the p ole gaugino masses in terms of the co ecients of the running

gaugino mass term op erator, M (  + h.c.), expanded as a series in the gauge couplings:

i i i

i i j

(1) (2)

M = M + M +


(8)

i

i ij

4 4 4 5

(1) (2)

where M are the known LO co ecients, and M are the NLO co ecients that wewant ultimately to extract.

i ij

2

The p ole gaugino masses at O ( ) order are given by

z

 the contribution from the renormalization of the gaugino wave function. This coincides with the M

i

correction present also in the full theory.



 the contribution from the renormalization of the gauge couplings. This coincides with the M correction

i

present also in the full theory.

 a one lo op diagram (gauge correction to the gaugino propagator). As said, it is not dicult to see that

it gives the same contribution of the asymptotic expansion in the external gaugino momentum of the

two-lo op  diagram of g. 1.

More in detail the p ole gaugino masses in the e ective theory are

 

4 4

i i i j

(1) (2)

z  e

M 1+ C (G)( M + M + M : (9) M j = + g ) +

i i IR

i i p ole

i ij

4 4 " " 4 4

uv ir

A further simpli cation o ccurs. The sum of the three e ective-theory quantum corrections, all prop ortional

(1)

to M , is b oth infrared and ultraviolet convergent (b ecause the combination M = is RGE-invariant at one

i i

i

(1) (1)

^

lo op). For this reason we do not need to worry ab out the O (") terms that distinguish M from M .

i i

2.3 Matching

The matching pro cedure is particularly simple: the running gaugino masses in the e ective theory at NLO

order are simply given by the full theory result, omitting those quantum corrections that are present also in the

e ective theory result, eq. (9). The MSSM running gaugino masses at NLO order are

( " #

X X

F

n i i

M = T (R ) g (x )+ 2C (G)g (x )+ T (R )g (x ) +

i i n 1 n i 2 n i m T m

4 M 4

n

n m

)

(10)

 

3

2 2

X

 

j

n

0

+ C (X ) g (x ) 2x g + g (x ) : (x )ln

j n C n n  n n

1

2 2

4 M (4 )

n

j=1

The parameters are renormalized as discussed in sec. 2.1. The last term is the e ect of a p ossible Yukawa

2



coupling  SX X in the sup erp otential, where S is a gauge singlet . The functions g , g and g have b een

n n n 1 2 C

de ned in eq.s (4) and (6), while g and g are

T 

2

1 2x 3

2

ln(1 x ) ; (11 a ) g (x) =

T

2

6x 2

ln(1 + x) x 4

g (x) = [2x(3 + 2x) x ln (1 x) +(5+3x)ln(1+ x)] + Li (x)+(x !x): (11 b )

 2

3 2

2x x

The functions g and g are normalized such that g (0) = g (0) = 1, while g (0) = g (0) = g (0) = 0. So far

1 2 1 2 C T 



we have assumed that R = X  X . If there are also messenger elds  in a real representation R , the

n n n 

appropriate group factors are T (R )=T() and C (X ) ! C ().



In the limit x ! 0 the NLO prediction for the running gaugino masses do es not dep end on the messenger

n

sp ectrum

h i

X

2

F ()

F M

n i i

T (R ) 1+ 2C (G) at   M : (12) M () =

i n i n i

4 M 4

n

n

We rememb er that C (G)=f0;2;3g. This prediction for the three gaugino masses, without knowing the values

i

of the F =M parameters, is of interest only if their numb er is less than three. This happ ens, for example, if

n n



the messengers lie in a 5  5 representation of SU(5).

2

The computation of Yukawa corrections, that we do not present, involves one new feature: a renormalization of F =M . For

n n

simplicity,wehave given the Yukawa correction assuming that all the F and M are pro duced byavacuum exp ectation value of

n n

the sup er eld S . In this case the new feature b ecomes an irrelevant common renormalization of F =M . The corresp onding e ect

n n

dr ) renormalization of F . in the nal result, eq. (10), has b een absorb ed in an appropriate (non

n 6 1.1 detectable LSP decay invisible LSP decay dangerous LSP decay 1.08 Unified SuGra Gauge-mediation j

ρ 1.06 with (n5, n10) = (1,0) / i ρ ρ ρ 1.04 1 / 2 ρ ρ 1 / 3 ρ ρ 1.02 2 / 3

1 Measurable ratios 0.98

0.96

104 105 106 107 108 109 1010 1011 1012 1013 1014 1015

unified messenger scale in GeV

Figure 2: NLO predictions for the measurable ratios  (M )= (M ) (  M = ) in the minimal gauge

i Z j Z i i i

mediated model (n =1, n =0) for generic values of the uni ed messenger mass. For comparison, we also

5 10

show the prediction of uni ed supergravity models (neglecting possible smal l GUT-scale e ects).

3 Predictions of uni ed messenger mo dels

We will show the NLO predictions in mo dels where the messenger sp ectrum satis es uni cation relations. These

mo dels are not only more predictive but also more app ealing: the successful uni cation of the gauge couplings is

not destroyed and a uni ed messenger sp ectrum helps in avoiding undesired one-lo op contributions to sfermion



10 masses. Tobe more sp eci c we assume that the messengers ll n copies of 5  5 and n copies of 10 

5 10

representations of the uni ed group SU(5), so that the messenger contribution to the one lo op co ecientofthe

gauge functions is T (R )=n +3n . We assume that the messenger mass parameters M and F arise from

i 5 10 n n

the vacuum exp ectation value of one SU(5)-singlet eld S coupled to the messengers via Yukawainteractions



dr mass parameters at   M are thus obtained  SX X . Imp osing the uni cation relations the running

n n n n

with NLO precision via two-lo op RGE evolution from M down to :

GUT

F

n

=U (M ! )[x (M )+ x ] for messengers R uni ed in R x ()=

n GUT N GUT n N n N n

2

M

n

where  x represent unknown one-lo op threshold e ects at the uni ed scale, that we will neglect. `Reasonable'

n N

threshold e ects give small corrections also when x  1. We also neglect NLO Yukawa corrections, p ossibly

relevant when x  1. The NLO RGE equations for M and F are given in app endix B (the combination

n n

F =M is RGE-invariant, and is not corrected by threshold e ects). Already at LO, the RGE evolution of the

n n

2

messenger sp ectrum gives corrections of relative order O (x ln M =M ) to the relations M / : for light

GUT i i

messengers the leptonic messengers are approximately 2 times lighter than the hadronic ones. This explains the

3

larger e ect present for light messengers (x  1) .

N

Fig. 2 shows the prediction of the minimal mo del with (n ;n )=(1;0) for the measurable ratios  = (the

5 10 i j

exp erimental errors on the gauge couplings negligibly a ect the predictions for the  (M )  M (M )= (M )).

i Z i Z i Z

3

Values of x > 1 (negative squared tree-level masses for scalar messengers) give an acceptable sp ectrum of physical messenger

N

masses (x () < 1), leaving only charge and/or colour breaking (CCB) minima at very large eld values. This situation is not

n

forbidden. On the contrary, non-dangerous CCB minima are quite generic in gauge mediated mo dels [14 ]. 7 1.15 (1,0) (0,1) (2,0) (1,1) (3,0) (2,1) 1.1 (4,0) j

ρ (5,0) / i (6,0) = (n , n )

ρ 5 10 1.05

ρ ρ 1 / 2 1 ρ ρ 2 / 3 Measurable ratios 0.95

0.9 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 10 100 1000

unified messenger scale in TeV

Figure 3: As in g. 2, but with a `less minimal' messenger content.

In this gure wehave considered the whole range of p ossible messenger masses, distinguishing the smaller values

of the messenger mass for which the lightest sup ersymmetric particle (LSP) decays b efore escaping detection,

from the higher values for which the LSP decayissoslow (in absence of R -parity breaking) that destroys the

nucleosynthesys pro ducts [3].

In g. 3 we consider mo dels with more than a single uni ed messenger. For the sake of illustration wehave

combined their contributions assuming that all the messengers have the same uni ed M and F . For values

n n

of M higher than the ones considered in g. 3 (x  1) all neglected NLO terms are completely irrelevant, the

prediction do es not dep end on the messenger content, and is the same as in g. 2. In all plots wehave xed

the F -terms requiring that the running gluino mass b e M (M ) = 500 GeV. Any other reasonable value of the

3 Z

gluino mass gives the same prediction for  . We rememb er that wehavenot included the one-lo op corrections at

i

the electroweak scale to M , that dep end on unmeasured (but measurable) and unpredicted parameters (mainly

i

the -term). The error on these predictions, due to remaining NNLO e ects, is estimated to b e at the per-mil le

level, much smaller than the exp ected exp erimental error on the gaugino masses.

In g.s 2, 3 wehave also plotted the corresp onding NLO prediction of uni ed sup ergravity mo dels, without

including unknown p ossible GUT-scale corrections. The uni cation relation  / 1I is infact only corrected

i

at the % level by GUT-scale threshold [7] and gravitational [15] e ects, that could instead give much larger

corrections to the uni cation relations for the and for the M . The RGE contribution from the top A term,

i i

driven towards its IR xed p ointvalue, A (M )  2M , is also numerically negligible [6]. The same can b e said

t Z 2

for the b ottom and  contributions, that remain negligible also if tan is large.

At this p oint it is also interesting to discuss the NLO correction that the presence of messenger elds gives

to the uni cation prediction for (M ). The p ercentage correction to (M ) is plotted in g. 4 for mo dels

3 Z 3 Z

with di erent messenger content, assuming that all di erent messengers havea common uni ed value of the

2

M and F parameters (if F  M it is only necessary to assume that the various M have the same order of

n

magnitude). If n +3n > 5 to o light messengers give a non-p erturbativevalue of the uni ed gauge coupling.

5 10

It is interesting to see more in detail why this correction is much smaller than its nave exp ectation and whyit

exhibits some curious prop erty. For given values of the uni cation scale and of the uni ed gauge coupling, the low

1

th RGE

(M )= +  . energy gauge couplings receivetwo contributions due to the presence of messengers: 

Z

i i

i 8 ) Z 5% M ( 3 α (1,0) = (n5, n10) 4 n5, + 3n10 = (2,0) 6 (3,0) (0,1) (4,0) (1,1) 3 (5,0) (2,1) (6,0) (3,1) (0,2) 5 2 4 1 3 2 1 0

-1 Percentage correction to the prediction for 104 105 106 107 108 109 1010 1011 1012

Unified messenger mass in GeV

Figure 4: Percentage correction to the uni cation prediction for (M ) due to the presence of messengers

3 Z

uni ed in the simplest representations of SU (5).

th

The rst contribution,  , is due to messenger thresholds (gauge corrections distort the uni ed messenger

i

sp ectrum); the second one is due to messenger corrections to the running of the gauge couplings (we can

reabsorb the one-lo op contribution in the de nition of the uni ed gauge coupling, and consider only the two-

lo op contribution). In a suciently accurate approximation the two corrections are

n o

X

1

(2) (2)MSSM

MSSM RGE

b ln r b ln r  (13 a ) =

j

j i

ij ij

4

j

2

X X X X

2

1  1

F M

p

th

= b ln 4T (R )C (X )lnr (13 b )  =

i n j n j

i

i

2

4 M 4

p

n n

j

p2R

n

P

p

where extends over all the fermionic and scalar messengers, b is the contribution of a given messenger

i

p

P

(1) (1) (2)

p

MSSM

to b = b + b , and b and b are the one and two-lo op co ecients of the gauge -functions in

i

i i i ij

p

2 2 1=b

i

> 1 presence of messengers (explicitly given in app endix B). We de ne r  (1 b ()=(4 )lnM = )

i i i

GUT

MSSM

while r is its value without messengers. The overall correction to the uni cation prediction for the strong

i

coupling constantis

mess th RGE 2 th RGE

(M )( +  ) (14)  (M )= +  =

Z i 3 Z 3 3

3 i i

th RGE

where  = f5=7; 12=7; 1g. The two corrections,  and  can be quite large (O (0  20)%) and

i 3 3

dep end separately on n and on n . However the sum of the two contributions, plotted in g. 4, is much

5 10

smal ler, typically p ositive,  = (0  3)%, and dep ends only on the correction to the one-lo op -function

3

mess

co ecient b = n +3n (this is not true in the limit x  1, where sup ersymmetry-breaking e ects b ecome

5 10

i

th

relevant). This cancellation can b e seen summing the expression for  in the limit F  M , eq. (13 b ) (in which

i

wehave inserted the messenger masses M obtained via one-lo op RGE evolution), with the RGE correction (in

n

whichwe insert the general values of the two-lo op -function co ecients, written in eq. (B.2)):

( )

2

1 r

F M

j

(2)MSSM

RGE mess th mess

= ln r + b ln +  2C (G )b : (15) =  

i i

i i i i

ij

MSSM

4

r

j 9

Corrections due to p ossible messenger Yukawa couplings would cancel out. These cancellations repro duce

the exact result found by J. Hisano and M. Shifman in [9, 10] working in toy mo dels with the holomorphic

sup ersymmetric gauge couplings. As shown in [16] this same reason is at the basis of the analogous cancellation

between RGE and threshold e ects encountered in our NLO computation of gaugino masses at F  M .

4 Conclusion

We have computed the next-to-leading order corrections to gaugino masses in gauge-mediated mo dels for

generic values of the messenger masses M . In uni ed messenger mo dels there are up to 10% corrections to

the uni cation-like relations M () / ()between the running gaugino masses and the gauge couplings, but

i i

only if the messengers are strongly splitted by sup ersymmetry breaking. If instead M>100 TeV there are only

small (few %) corrections to the leading-order approximation M / , as shown in g.s 2 and 3.

i i

Wehave also studied the messenger corrections to gauge coupling uni cation. As a result of cancellations,

dictated by sup ersymmetry, between large RGE and threshold corrections the predicted value of the strong

coupling constantistypically only negligibly increased, as shown in g 4.

2

In the limit M  F (heavy messengers) the same NLO prediction for gaugino masses, together with NLO

predictions for sfermion masses, can be obtained [16] combining the techniques describ ed in [17] and [9, 10].

2

If F  M it is more dicult to obtain a NLO prediction of sfermion masses; however the LO results [11]

show that the e ects of large sup ersymmetry breaking in the messenger sp ectrum are much less relevantinthe

sfermion sector than in the gaugino sector.

Acknowledgements We are grateful to Riccardo Barbieri for many discussions, and to Riccardo Rattazzi for having

p ointed out a missing term (!) in the renormalization of our result.

A Relevant Lagrangian in quadri-spinor notation

We consider a theory with messenger chiral sup er elds  and  in self-conjugate complex representations of the SM

L R

T

gauge group (with generators T and T ). In presence of the sup erp otential

Z

2

^

W = d S  ; hSi=M+F:

L R

p

2  +


contain the following mass eigenstates: messenger fermions , with The messenger sup er elds  = A +

L R

p

 2 2 2

a Dirac mass M , and pseudoscalar (scalar) messengers A  (A  A )= 2 with mass M = M  F  M (1  x).

 L



R

The sup ersymmetric gaugino Lagrangian is

p

m

 y T  T T

      

L =  iD  ( + ) 2g (A T  +  TA  TA A T )

 L L L



R R R R

L

2

where D is the standard gauge-covariant derivative and , and are Weyl fermions with the same chirality.

 L R

Wewant to rewrite the Lagrangian in terms of mass eigenstates. In order to employ our Mathematica [18 ] co de for

analytic computation of Feynman graphs, we need to write the messenger fermions as Dirac quadri-spinors and the

gauginos  as Ma jorana spinors :

   



L

c T



; = = C 

T







R

where C is the charge-conjugation matrix. The gaugino Lagrangian b ecomes

1

y y

    

L = (iD= m) g (A T  + TA A  + TA ) (A.1)

5 5 +



+

2

and the D terms b ecome

a  a a   a  a

D = A T A A T A +


= A T A + A T A +

L R +

L R + 10

  

The gaugino  propagator is a standard fermion propagator. It is p ossible to show that graphs with ,  and

 

 propagators have the same value of the standard `', by appropriately rewriting the vertices in terms of charge-

4

conjugated elds . This shows that the gaugino can b e treated as an ordinary fermion eld, but with symmetry factors

computed like the ones of a real eld.

2

In a general theory there will b e several messenger pairs, each one with its M , F and x  F =M .

n n n n

n

B RGE evolution

The necessary RGE can b e read from the literature [6 ]. The RGE equations for gauge couplings and gaugino masses in

the dr scheme are

" #

X X

d 1 1

(1) (2) (2)

2 2

= b b + g b 

j a

i ij ia

2

2

dt g (4 )

i

j a

" #

2

X X

g d

(1) (2) (2)

2 2 2

i

M = g b b M + g (M + M )+ b  (M A )

i i i j i a

i j a

i ij ia

2

dt (4 )

j a

2 2 2

where t(E )  (4 ) ln M =E and a runs over the third generation particles, a = ft; b;  g. In a general sup ersym-

GUT

metric mo del with fg matter sup er elds the co ecients are

X X

(1) (2) (1)

b = 3C (G )+ T (); b =2C(G )b  +4 T ()C (): (B.2)

i i i ij i j

i ij i

 



In the mo dels under consideration the eld content is given by the MSSM elds plus n copies of 5  5 and n copies

5 10

10 SU(5) messenger multiplets (in the e ective theory without messengers, the co ecients are obtained taking of 10 

n = n = 0). The values of the co ecients are

5 10

a : t b 

1 0

! !

1 33=5

26=5 14=5 18=5

(2) (1)

A @

6 6 2 1 b = 1 +(n +3n ) b =

5 10

ia i

4 4 0 1 3

!

7 n 23 n 9 n 3 n 32 n 48 n

199 27 88

5 10 5 10 5 10

+ + + + + +

25 15 5 5 5 5 5 15 5

(2)

3 n n

9

5 10

b = : + + 25 + 7 n +21n 24 + 16 n

5 10 10

ij

5 5 5

4 n 6 n 34 n

11

5 10 5

+ + 9+6n 14 + +34n

10 10

5 15 5 3

In numerical computations it is useful to employ the RGE for   M = , since it starts at two-lo op order

i i i

" #

2

X X

d g

(2) (2)

2 2

i

 = b g M + b  A : (B.3)

i j a

j a

ij ia

dt 4

j a

2

When F  M sup ersymmetry is hardly broken in the e ective theory b elow the messenger scale: the couplings at

the sup ergauge gaugino vertices di er (bya numerically negligible amount) from the corresp onding gauge couplings.

However in this case the messengers are light so that it is sucient to employ the one lo op RGE equations b elow the

messenger scale.

Finally the gauge contribution to the 2 lo op dr RGE equations for the sup ersymmetric messenger masses M and

n

for the `F -terms' F is

n

n o

d d 1

(1)

2 2 4 2

g g 2C (X )2C (X )+g 2C (X )b : ln M = ln F =2C (X )g

i n j n i n n n i n

i j i i

i

2

dt dt (4 )

g

4

~

Only the X X diagram of g. 1, that needs a Ma jorana gaugino propagator to b e non-zero, requires a more detailed treatment

of the charge conjugation factors. 11

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