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Renormalizations in softly broken SUSY gauge theories
ELSEMER
Nuclear Physics B 510 (1998) 289-312
Renormalizations in softly broken SUSY gauge theories L.V. Avdeev
l,
D.I. Kazakov 2, I.N. Kondrashuk 3
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, I41 980 Dubna, Moscow Region, Russia
Received 10 October 1997; accepted 27 October 1997
Abstract The supergraph technique for calculations in supersymmetric gauge theories where supersymmetry is broken in a “soft” way (without introducing quadratic divergencies) is reviewed. By introducing an external spurion field the set of Feynman rules is formulated and explicit connections between the UV counterterms of a softly broken and rigid SUSY theories are found. It is shown that the renormalization constants of softly broken SUSY gauge theory also become external superlields depending on the spurion field. Their explicit form repeats that of the constants of a rigid theory with the redefinition of the couplings. The method allows us to reproduce all known results on the renormalization of soft couplings and masses in a softly broken theory. As an example the renormalization group functions for soft couplings and masses in the Minimal Supersymmetric Standard Model up to the three-loop level are calculated. @ 1998 Elsevier Science B.V. PACS: ll.lO.Gh; ll.lO.Hi; 11.3O.Pb Keywords: Soft supersymmetry breaking; Spurion; External superfield
1. Introduction Supersymmetry, if it exists, must be broken. The gauge theory with softly broken supersymmetry has been widely studied. To break supersymmetry without destroying the renormalization properties of SUSY theories, in particular the non-renormalization ’ E-mail avdeevl@thsunl .jinr.dubna.su. *E-mail [email protected]. 3 Email ikond@ thsun 1.jinr.dubna.su. 0550-3213/98/$19,00 @ 1998 Elsevier Science B.V. All rights reserved. PIISO550-3213(97)00706-2
theorems
and the cancellation
called soft terms
of quadratic
[ 1] , They include
divergencies,
the bilinear
one has to introduce
and trilinear
scalar couplings
mass terms for scalars and gauginos. A powerful method for studying SWSY theories which keeps s~pers~mm~try technique
[2,3].
the SQand the rna~~~est
It is also a plicable to softly hroke~ SWSY mode [ 1,4,S] AAs has been shown by Yamada [ 6 j
by using the “spuri~n~’ external s~~r~e~ds
with the help of the s~urio~ method the ca~cnlation breaking
terms
is a much
simpler
of the j3 f~~ct~on~ of soft SUSY-
task than in the component
approach.
Using
the
spurion technique he has derived very efficient rules which allow him to calculate the soft term divergencies starting from those of a rigid theory in low orders of perturbation theory. Hvwever, these rules need some modification since, as mentioned by Yamada 161, there might appear some problems in t e calcufation of vector vertices because they ~~~~~j~ cl-iii-al derivatives acting on external lines. In the present paper we develop such a modification of the spurion technique in gauge theories which takes care of the above mentioned problem. We formulate the Feynman rules and show that the ultraviolet divergent parts of the Green functions of a softly broken SUSY gauge theory are proportional to those of a rigid theory with the spurio~ fields being factorized. In genera], we consider a softly hroke~ s~~ersymmetric barge theory as a rigid SUSY theory ernb~~~ into the external space-time i~de~nde~t s~per~eld~ so that all the parameters as the couplings and masses become external superfields. This approach to a softly broken sypersymmetric theory allows us to use remarkable mathematical properties of N = 1 SUSY theories sucfi as Nan-reno~a~~zat~on theorems, cancellation of quad~t~c d~ver~e~ces~ etc. We show that the reno~a~~zat~on procedure in a softly broken SUSY gauge theory can be perfo~ed in exactly the same way as in a rigid theory with the reno~alizatju~ constants being external super~elds. They are related to the co~es~ond~~~ reno~ali~ation constants of a rigid theory by the coupling constants redefinition.
This allows us to find explicit
relations
between
the reno~ali~atio~s
of
soft and rigid couplings. Throughout the paper we assume the existence of some gauge and SUSY invariant re~u~~izatio~ and a minjma~ suhtractio~ procedure. Though it is some problem by itself, we do not consider it here. Provided the rigid theory is well defined, we consider the mod~~cations which appear due to the presence of soft SUSY breaking terms. To be more precise, in the following sections when discussing one, two and three loop calcn~at~o~s of the reno~alization constants we have in mind dimensional reduction and the minimal subtraction scheme. Though dimensional reduction is not self-consistent in general, it is safe to use it in low orders and all the actual calculations are performed in the framework of dimensional reduction [7-IO]. nevertheless, our main fo~ulae are valid within any jny~r~ant regular~zat~on, which keeps the action unchanged, and a minimal subtraction scheme.
291
2, Softly broken pun! N = 1 SUSY Yang-Mills
theory
Let us first consider pure N = 1 SUSY Yang-Mills The Lagrang~an of a rigid theory is given by
Here W” is the field strength chiral superfield
theory with a simple gauge group.
defined by
W”= o2 (e -VP%?“), where V, = VA (~A)~ and TA are the generators of the gauge group C. To perform a soft SUSY breaking, one can introduce a ga~gi~o mass term. This is the only term allowed in a pure gauge theory which does not break the gauge ~nv~i~ce. The soft SUSY breaking
term is
where /t is the gaugino field. To rewrite it in terms of superfields, let us introduce an external spurion superfield q = @*, where B is a ~r~smannian parameter. The softly broken ~grangia~ L soft=
can now be written as
s
’
d2B~(1
-2pf12)TrW”W,+/d26-${l
-2ji,t?)Tr#‘*Wa.
(2)
In terms of component fields the interaction with external spurion superfield leads to a gaugino mass equal to mu = p, while the gauge field remains massless. This external chirat superfield emerging
can be considered
from su~r~~av~ty,
as a vacuum exp~tation
value of a dilaton superfield
however, this is not relevant to further ~~ns~derat~o~.
Consider now the Feynman rules correspdnding to the Lagrangian (2). For this purpose it is useful to rewrite the integral over the chiral superspace in (2) in terms of an integral over the whole superspace.
t-l4s2 (1 Consider
(3)
> part of the action
d4x 80 d28Tr -& s
.
2p~*)Tre-“b”e”D2e-“LS,eV
the quadratic
+
One has
(1 - 2~8”) B’V( x, 8, #)D2DkV(
x, 8,8>.
To find the propagator, fixing condition
one first has to fix the gauge. In a rigid theory the gauge
is usually
D2v = f,
taken as
LFV = J;.
This is ~uivalent
(4)
to adding to the action ( I> a gauge-non-inv~i~~t
term propurtio~al
to
c gauge-fixing - -Tr By joining
dff
(ff
this together the quadratic
1t can also be rewritten 4
= -Tr &
88
+ jr).
(5)
part of the rigid action takes the form
as
d%TrV
[L-l,,? + $1
d%
s’ s where we have introduced
[ 111
the projectors
The inverse operator then gives the vector propagator (v”(e,)vqzz)).@
= ;flt~2;~?3@qZ,
The next task related
-Z2)@.
to the determination
of the vector propagator
asso&~ated ghost term. Under the gauge tra~sfo~atio~
is to find the
the sup~~~e~d V is transfo~ed
as
or in the infinitesimal
form
OL
v+V+H(V)R+R(V)A=V+ ~o~se~ue~tly,
c ghost =
the ~so~~ated
s
=
-p
(H(V)c+A(V)E~
d28d28
(b+t;)
where b and c For a system in order to take The reason for
Tr;
(A + A) + coth
(g)A (n-n)].
ghost term is
d2@Tr$bD2
s
A
-f O[
+/d28Tr$6L?’ A (c+C)
(N(V)c+l?(V)L;)
+coth
are the ~add~v-Popov ghost chit-al superfields. in the external spurion field we can change the gauge fixing condition (4) into account the interaction of the associated ghost fields with the spurion. this is the appearance of these terms in due course of renormalization.
Here, we depart from the Feynman
rules of Ref. [ 61. Our aim is also to get the same
common spurion factor for the ghost propagator as for the vector one, as it will be clear below. For these purposes we choose the gauge-fixing condition in the form
or
where we have introduced
the notation
22 = g2 (I + #iLiP+ /.&a2+ ~~=~~~~) . Note that due to the ~rassma~nia~
Now the gauge-fixing
origin of B
term can be written in the standard form
t be a real ~-independent
which leads to the following
quadratic
s~pe~~~~d. l[n what follows we take
part of the action in a softly broken theory
One now has to find the inverse operator, It can be done analogously
to the rigid case
by introducing the projection operators. One should take into account, however, that while integrating the ~o~a~a~t derivatives by part they may also act on the sp~rjon field. Strictly
speaking,
this leads to additional
terms in the propagator
which are, however,
suppressed by powers of momenta and are iness~ntial in the analysts of UV diverge~~es~ To understand it better, one can treat the spurion terms in E&q_C101 as inte~a~tio~s~ and to find the propagator, one has to take into account an infinite chain of this kind of insertions adding up into a geometrical progression. It can be done by the method of Ref. [ 5 1. If in due course of this procedure some of the covariant derivatives act on a spurion field, one does not get enough powers of momenta to cancel the denumi~ato~ and the resulting terms are thus suppresse for a high momentum square. Thus, as it has been also noted in Ref. [6], for the calculation of the divergent parts of the Green functions it is suf~~~e~t to take the part of the vector propagator where the chiral derivatives do not act on spurions.
Having this in mind, one can proceed in full analogy with tbe rigid theory, replacing the coupling
g* by g2, and get
where by i~eleva~t
terms we mean the ones d~reasing
faster than l/p2
Thus, we get a simple relation between the vector propagators
+irrelevant
for large y2.
of soft and rigid theories:
terms.
Notice that control to Ref. 161 this relation is valid for any choice of Q. The change of the gauge fixing condition leads to the change of the ghost Lagrangian
(6). It becomes
As one can see from Eq. ( 12), the situation with the ghost propagator that contain ghost superfields derivatives in the ghost-vector
and the vertices
is more simple. This is due to the absence of chiral interaction. Nevertheless, when calculating the inverse
operator one has to repeat the same proc~ure with the cov~ia~t derivatives arising from the source terms. In complete analogy with the case of the vector propagator the essential derivatives
part of the ghost propagator do not act on the spurion
powers of momenta. (G(z~)~(z~))~~~~
is obtained
by ass~mjng
that the covariant
field. The other terms are suppressed
by the
I-Ience, one again has = @‘/g’)
(C(zj
)G(II))~~~~~
+
irrelevant
terms,
(13)
where G stands for any ghost superfield. Thus, to perform the analysis of the divergent part of the diagrams in a soft theory, one has to use the same propagators as in a rigid theory multiplied by the factor g*/‘g’. It is also obviously true for any vertex of the ghost-vector interactions of the softly broken theory ( 12). Each vertex of this type has to be multiplied by the inverse factor g2/g? The situation is less obvious with the vector vertices. Here, one has two terms as in Pq. (2) which differ only in the order of cov~iant derivatives, one being complex conjugates to the other. Per~o~ing ~rassmannia~ algebra in any diagram one can always replace one term by the other; so actually one has only one term. This is what we have in a rigid theory. In a soft theory the situation is similar but the first term is
multiplied by a factor (1 - 2pe2) , while the second by the complex conjugated one (1 - 2~6~). Hence, one can consider the vector vertices of only one type, as in a rigid theory, multiplied by a factor (1 - ~8~ - pe2), i.e. by g2/@ as in the case of the ghost-v~tor vertices. Thus, we see that any element of the Feynm~n rules fur a softly broken theory the ~o~es~ondi~g element of a rigid theory factor which is a ~olynomi~ in the Grossman ~oord~nates~
2.2. The ultraviolet
counretierms
Consider now the Green functions of a softiy broken theory ~~nstru~t~ with the help of the Feynman rules delved in the previous section. Due to the presence of a factor g2fg” in the vertices and of the inverse one in the propagators, any ~eynma~ d~a~rarn in a broken theory is given y that of a rigid theory with the spurion factors on the lines and vertices. Our aim is to show that these polynomials in grassmann variables can freely flow through the diagram and are finally collected in front of a diagram as a common factor. Indeed, consider t ~o~es~o~di~g ~rassrna~n~~ ~~tegraiio~~The usual pro~~~~e of evaluation of Grassma~n~~ integrals in a rigid theory is the following: 63 If there are no chiral derivatives II, or ija on the line, it is propo~ona~ to the ~-function and can be shrunk to a point. (ii) If there is some number of derivatives, one can remove them from a line integrating by parts. (iii) Repeating this procedure several times and using the prosodies of chiral derivatives (D,, Da) = {&, II,) = 0,
D,D@ = r,@,
D2B”D2 = 16pW,
etc.
to reduce their number one can finally come to a final integration where all the chiral derivatives are concentrated on one line. The powers of momenta which appear in due course af this procedure cancel some propagator lines so that the resulting diagram diverges. The final integration is ~rfo~ed with the help of the relation
Consider what happens if one adds Grassman~i~n polynomials en the lines and vertices. Suppose first that all chiral derivatives inside a graph are contracted and integrated inside the graph so that there are no external derivatives left. If one of these internal derivatives acts on a s~urio~ field it creates additional 8” instep of II”,. Then either n survives till the last integration, or some other D, acts on it. Thus, at the last step of integration one has instead of ( 14) either
It is easy to see that both the expressions
are zero. Thus, the configuration
chiral derivative acts on a spurion field gives no contribution.
field flows through the diagram freely and can be factorized. Consider now the case when a graph contains some derivatives lines when perfo~in~
the Grassmannia~
when the
This means that the spurion that act on external
i~te~rat~on~ This case ~o~esponds
to vector
interactions. Let some covariant
derivative,
which originally
was inside the diagram,
pass through
the diagram during the above mentioned procedure and act on the external vector line. We can trace this derivative in the diagram of the soft theory. On its way through the diagram
it can meet some “spurion”
factor in a vertex or a propagator.
Integrating
by
parts one has two terms. The first one, when the chiral derivative acts on a spurion, and the second one, when it acts on the nei~~bouri~g propagator. When the ~ov~ja~t derivative does not act on a sp~r~on it goes through the diagram like in a rigid theory and acts on external line. As a result one gets the expression g2 fDLYVMz). In the other case, when the covariant derivative acts on the spurion, the number of covariant derivatives is reduced compared to the rigid theory. Then, either one does not have enough of them to cancel the powers of momenta in the denominator or to get the non-zero answer or one has the same diagram as in the rigid case, but with the factor Dg*, that goes outside the diagram. Independently of the position of this factor inside the diagram, the nal expression one gets after the ~al~~~atio~ of the diagram contains ~~*~) V’“’which together with the usual contribution, gives II, (j2V’“) instead of g”D,V”’ in the rigid case. All together these factors S2, which are met by the chiral derivative on its way to the external line of the diagram, are collected in a monomial R(S2), Therefore, the final expression for the diagram contains D, (R(S2) V”). The same conclusion is true for all derivatives of t e divergent vector dia~r~nl that act on the external lines. Since the cou~terterms in a given order of perturbation theory should be gauge invariant, in the final expression only the countertetms with four covariant derivatives survive. The have the form
where Rr , R2, R3 and F&tare some monomials in $. This expression witk the ~o~es~on~~~~ ~ounterte~ of the rigid theory
should be compared
R(g2)V~rD”V”L52VkD,V’, where
The momentum integral for the diagram in a soft theory is identical to &at in a rigid one since the algebraic operations with the covariant derivatives coincide for the both cases.
We now want to argue that the monomials
Ri(g2) can stand anywhere in Eq. (IS),
i.e.
that the spurions can flow through the diagram. Indeed, in a rigid theory the counterterms are absorbed
into the redefinition
of the field V and the coupling
g2
vs = z~~2(g2~v) & = qg2>g2* In a soft theory Ri are the functions malization
constants
become
ctf the external
these functions
sp~rion s~~~~el~
too. Collecting
type (15)) one should get an expression which being quantities obeys the gauge invariance, namely
and the renor-
alt counterterms
written
of the
in terms of the bare
where the primed rno~orn~~s mean that pat-Q they are absorbed into the ~e~o~aIization of the field Vs. Now it is clear that the covariant
derivatives
in Eq. (16)
do not act on monomials
Ri(g2>. Indeed, if one of the covariant derivatives D, in Eq. ( 16) acts on the monomials Ri or R& the two exponentials cancel each other and one is left with Tr W” that is equal to zero. In the case when both the chiral derivatives in Eq. (16> act on Ri and Ri in both cases the expo~e~tials cancel and one has a divergent constant that should be absorbed into the reno~ali~tio~ of the vacuu density. Hence, one can take the monomials R$ and Ri out of the cov~ia~t derivatives D,. The other derivative, B2, does not act on the monomial for the reason that e-P4D”eG*82 = 0 so that in the bracket in Eq. ( 16) only the chiral spurion q = O2 survives and fi2 does not act on it. Thus, we come to a conclusion that the monomials can be factorised in front of the expression
in Eq. ( 16) just like in the rigid theory and the only difference
rigid and softly broken theories is that the coupling gz9 i.e.
constant
2
between
the
should be replaced by
Zi = Zj (g* + g”] *
3. Softly broken SUSY gauge theory with chiral matter Consider
now a rigid SIJSY gauge theory with chiral matter. The ~agrangian
in terms of s~pe~elds Lrigid
d28d28&‘(eV)i@j
=
written
looks like + /dZOV+~c.i26+i.J,
(17)
s
where the superpotential
W in a general form is4
W = ~~~~k~~~j~~ + ~~ij~~~j.
(181
4 To avoid further c~rn~l~c~~~~s (see, e.g. Ref. 161) we do not introduce linear terms in Eqs. ( 18), ( 19) s This corresponds to the case when none of the fields goes into vacuum. In the absence of singlets this is guaranteed by the gauge invariance imposing some restrictions on the couplings.
The SUSY breaking terms which satisfy the requirement
of “‘softness” can be written as
Like in the case of a pure gauge theory the soft terms (19) can be written down in terms of superfields
by usin
the external
spurion
field. Thus, the full Lagrangian
for
the softly broken theory can be written as
The ~agra~gia~
(~~~ arrows one to write down the ~ey~rnan rules fur the matted field
propagators and vertices in a soft theory. We start with the ~r~pa~at~r of the chiral field. For the purpose of the analysis of diver~~~~es ane can jgn~re the mass terms M’s and Bli, since in the minimal scheme they do not ~~~trib~te to the UV divergences= Then, the quadratic part looks Iike
The inverse operator is easy to obtain due to the nilp0tent
~ha~a~te~ of the sprain
fields.
One easily gets
$-irrelevant
terms,
ere tbe ~~~le~a~t terms again arise f~~rn c~variant derivatives
acting on spurions and
am suppressed by ~~~~~s of momenta. The vender-mattes vertices, a~~~rdi~g to Eq, (ZO), gain the factor (8: - {~~~~~~~ so that if in a dia~r~ one has an equal n~~~~r of chiral ~ro~a~at~rs and vector-u~att~r ~~rti~~s the spurion factors cancel. The chiral vertices of a soft theory, as it follows from Eq. (XI), are the same as in a rigid theory with the Yukawa couplings being replaced by i$, _ xi8 _ ~~jk~ f A_
;iij& + ;iijk - ~i~~~ .
The structure of the W counterterms in cbiral vertices is sirn~l~ to that of the vector vertices, but is simpler due to the absence of the covariant derivatives on external lines. This corresponds to the first case considered in the previous section. To get the UV divergent diagram, the ~ovar~ant derivatives should nut act on $~~r~~n fields, which
L.V Avdeev et al./Nuclear
means that sp~rions
factorize.
with the ehiral propagators, couplings
Physics 3 510 (1998) 289-312
In the diagrams,
299
when the chiral vertices are ~on~ra~t~
this results in the fo~~uwing effective ehange of the Yukawa
of a softly broken theory
Thus, the WV ~o~~terte~s
of a softly broken theory are obtained
from those of a
rigid one by the substitution
4. Renormalization of soft versus rigid theory: the general case The external
ma~izatio~
field ~o~st~~t~o~ ~es~r~~
of soft terms starting
above allows one to write down the renor-
from the known
without any new diagram calculation.
The following
reno~~~zat~on statement
of a rigid theory
is valid:
The statement Let a rigid theory (I), (18) be renormalized via introduction of the renormalization constants Zi, defined within same minimal subtraction massless scheme. Then, a softly broken theory (21, (20) is renormalized via introduction of the renormalization s~~e~e~ds 2~ whack are related to Zi by the ca~p~i~g ca~sta~ts r~d~~~it~a~
where the redejned couplings are
This allows us to find explicit relations between the renormalizations couplings
which is an explicit realization
of soft and rigid
on the level of final expressions
of the rules
A and I3 of Ref. [6]. The re~o~a~izatjon constants for vector s~~er~elds and gauge couplings are general s~per~el~s, i.e. ey depend on 71 and ji, white those for the chiral matter and ghost fields and the greeters of a supe~ote~t~~ are chiral supe~elds, i.e, they depend either on 71 or *. From Eqs. (24) and (25)-(27) it is possible to write down an explicit differential operator which has to be applied to the p functions of a rigid theory in order to get those for the soft terms, We first construct this operator in a general case and compare the resulting expressions with explicit calculations made up to two loops. Then, using
the formulated
algorithm
we calculate some three-loop
soft term p functions.
section, we consider some particmar models. To simplify the formulas hereafter we use the following
Consider
first the gauge couplings
In the last
notation:
ari. One has
wkere Zai is the product of the wave function
and vertex re~o~a~~zation
constants.
Though & and Zaj are general s~~er~e~ds, one has to consider only t since the Lagrangian ( 1) consists of two terms, a chiral and an antich~~al, and in each term only the proper chirality part contributes. Therefore, we consider the chiral part of Eq. (28)
aiding
over q one has
where the operator 131 ~xt~~c~s the hnear with respect Eqs. (25)-(27) the explicit form of Dr is
Combining
to 71 part of Z,i(Ly).
Due to
Eqs, (29) and (30) one gets
To find tke co~espo~d~n~ p functions one has to differentiate Eqs. (29) and (31) with respect to the scale factor bav~n~ in mini tbat the operator Dr is scale invariant~ This gives
ere yLyi is the logaritbm~c derivative of In Z&i equal to the a~orna~o~s dimension of the vector superfield in some particular gauges. This result is in complete co~es~ondence with that of Ref. [ 141, Indeed, from Eqs. (32) one can derive that the ratio ~~~/~~ is reno~al~zation group invariant.
301
4.2. Chiral matter Consider
now the chiral matter. Due to the non-renormalization
reno~alization
of a supe~orential.
the wave functions.
theorems, there is no
This means that the only reno~alization
The ~o~es~o~diag
term in the Lagr~gi~
comes from
in~l~d~~g reno~alizations
looks like
s
d20 $6 @g.j@ 1 .it
where the renormalization
(33) superfield
2; now has a decomposition
2; = &+& (6; + A~~~) z&,
(341
where the chiral( ant~chira~) reno~a~~zatio~ renormalizations
super-fields Zij ( Zi+k ) are the wave function
and .4f is the soft term (m2)j
Zj( &, jr, F) over the grassmann
renormalization.
To find, them expand
variables
-;f;;= 2; + Dt z$j + D, z;?j + DzZj~?j,
(35)
where the operator I)1 now has to take into account the q de~n~e~ce of the Yukawa ~ou~~j~g y “jk, D 1 is conjug ated to Dr and I)2 is a second order ~i~ere~t~al o~e~tor ich extracts the q+j dependence of 2;. They have the form D1 =mAiai-
a &2!j
+yabc-
d
ahbc
Substituting
+YbUC-
-
a
a
D] = mAiai_ - Aijkda; @ijk ’
(361
a
(371
@bnc
this into PEq. (34) one has
~~,i=Z~j+ i~j=Z~j
a
- A ijk dy”jk’
(Z$‘)~D,Zjk71=Z~k(Sjkf(Z-‘)f’D,Z/!rl), + B,Z~(Z~‘)jkd=
~~=(Z~~~~D~~~(Z~‘)~
CS~+ ii,Z~(Z-‘)fii)Z~j
(381 ))
(39)
- (Z~‘)lt>lZ~(Z-“>j;D~Z~(Z~‘)J”,
C4Q)
where Z~j satisfies Z~k Z~j = Z~ )
and is the square rout of Zi in the perturbative sense. The inverse one is (;?;‘>; = (8; - {Z-‘~~D~Z~~~~Z~‘~~. We can now write down the relations between the renormaliz of a superpotential. One has
(411
the bare couplings
(42) (43)
e last line can also
differentiating
wit
respect to a scale factor one gets after some alge
with L$ given by Eq, (40). ~iffer~~t~~t~n~ it with respect to a scde factor and using the explicit form of the seeon order d~ffc~ntja~ operator &, after some algebra one obtains a simple ~xp~~ssio~ for the /I function
Relations between the rigid and soft terms ~en~rrnali~t~~ns snbtractio~ scheme are surnrn~~~ as foli~ws;
in a m~ssless
minimal
L.I! Avdeev et al./Nuclear
The rigid terms
The soft terms
Pa, =
Pm,,
aiYa,
pjc, = L ( Milyi 2
1
+ j@#)
p;
1
= D
1Yni
= ; (&yli
+
f+jyj)
-(M’*D,d pijk ?
=
1( #jlyk 2
1
+
#lkyi
+
1
yijkyi)
px”
1
= 4 ( Aijryf
+ M’jDly()
+
-(yij’Dly; (Pm2>j =
a
- A ijk ayijk’ JCZi
&=DiO
+mi,ai& +$(m2)~(ynbc--& I d
a ‘%‘nbc
Ailk+
I
+
A’jkyi
1
>
+ yiikD,j I + y’jkD,yi)1
D2r;
d
D1 =mA;ai-
+hbc
303
Physics B 510 (1998) 289-312
+
Ybuc &‘bnc
+
d ybcn aYbcn
+ybnc--&+ybcn---&
>
5. Illustration To make the above formulae more clear and to demonstrate how they work in practice, we consider the renormalization group functions in a general theory up to two loops. We follow the notation of Ref. [8] except that our p functions are half of those of Ref. [8]. Note that all the calculations in Ref. [8] are performed in the framework of dimensional reduction and the MS scheme. The gauge ,6 functions and the anomalous dimensions of matter superfields in a massless scheme are the functions of dimensionless gauge and Yukawa couplings of a rigid theory. 5. I. One-loop renormalization In the one-loop order, the renormalization group functions simplicity, we consider the case of a single gauge coupling) y;‘) =cYQ, 7; (‘)=
of a rigid theory are (for
Q=T(R)-3C(G),
(51)
$y'k'yjkl -2&(R);,
where the Casimir operators T(W~AB=T~(RARB),
(52) are defined by C(G)~A~I
Using Eqs. (32) -( 50)) we construct terms
= ~ACD~B~D,
the renormalization
C(R);
= (R,~RA)~.
group functions
for the soft
L.c( Avdeev et al./Nuclear
304
Physics B 510 (1998) 289-312
pm ntn = a%@,
Pi (1) = $“(
(53) ;y+nrylk,, - 2aC( R);)
+M”( kAikn’yjknl + 2amAC( R){) + (i t-f j) , ijk (I)
PA
=
(54)
;Ai”( ;ykn”‘yln,,, - 2&(R);) +y’j’( $Ak”‘“yj,,,,, + 2amAC ( R) :) +(i cf j) + (i *
k) ,
(55)
- 4amf,C(R)j
[&2]1 (I) = iAik’Ajkl
f~ynk’(m2)~y,;kl
f iyik’(m2)gy&
+ iy’S’(m2):y,jk[.
(56)
One can easily see that the resulting formulae coincide with those of Ref. [ 81 with one exception: we have ignored hereafter all the tadpole terms assuming that they all are equal to zero because of the absence of linear terms in the action (see footnote 4). 5.2. Two-loop
renormalization
In two loops the situation y’*’ a =2a*C(G)Q y.Y2)= -(fn”‘yjnrr~
is more tricky. The rigid renormalizations
+ 2aC(R),y&)
Then, the soft renormalizations
r = dim G = aA~ , (57)
&@‘y;k[ - 2&(R)!),
- +(R);(
are
+ ~cz*QC(R)~.
( $y”k’y,>k[- 2aC(R)i)
are as follows:
2amA p,jt) = 4a*mAC (G) Q - -C(R)j( r
;yjk’yjkl - 2cuC(R)I’)
+~C(R);(IA’X’y~k,+2cumnC(R):), p;
12) = +“(yjkyy[kn
(59)
+2aC(R):‘Gj;)(;y”“‘y,,,,
-Mi’(AjkPylk,
- 2cxC(R);)
- 2am~C(R)I;@,)(~y”“‘y,,.s,
-kf”(yikpy[k17 +2aC(R):‘6~)(~A”“‘y,,,~, +B”a*QC(
+ 2cuC( R):‘$)
+AlJ’a2QC (R):
++Y””
j) + (in
Atk”A,jkn
+
(m”)
;Yjm
.$
(60) R);)
- 4y’*‘cx*QC (R):mA
-yij’(yknl”ylnln + 2aC( R)y#)
(*I = -(
+2amAC(R)i)
( +y”Sfyl,,,, - 2d(
-yij’ ( Ak”‘pyl,l,,7- 2cum,&(R):‘S:)(
[&I;
-2&(R);)
R){ - 4Mi’cu2QC( R)im,., + (i tf j),
Pi? (*I = -;Aii’(yk”‘f’yl,,,,
+(iw
(58)
;y”S’y,,,Yt- 2cuC(R);)
( iA”S’y[,,Yt+ 2amAC( R);)
k), (m2)&‘kpyjkn
+
+Yik”
(61) +
iyik”ylk,,
(m2)fyJk,, +
(m*)j
iyikp
(m*
),“lY.jk~
305
C~mp~ing these formulae with those of Ref. 181 we find that they coincide with one ization of m2 In Eq. (16) uf Ref. [8] has two extra terms, exception. The resow ~~4~~~ “,5 and the term ~ropo~ion~ to the mass of the s~-c~~ed e-scalars, $8. The first term, though it is not ~roport~o~~ to e2, still has tbe same origin. It has appear as a result of the s-scalar mass co~nte~erm in one loop. The presence or absence of these terms is ~eno~a~~~tio~
scheme dependents
The
version of dimensional reduction adopted in Ref. [ 81 ~o~es~o~~s to d~agr~ by diagram minimal subtractive of divergences and naturally incI~des the ~-scalar mass ~ou~ter~e~~
being formulated in the su~er~eld fo~alism no surprise that the co~es~o~di~g terms do not appear.
Our substitution
does not contain e-scalars;
There are two ways to resolve the noticed dis~r~p~~y. Either to r~e~ne the rebornovation scheme in such a way that these terms are not present, like the one in Ref. [ 121 I or to rn~~fy the dis~~sson
the ~uper~eld re~o~a~ization of this point in Ref. [ 151).
procedure to reproduce these terms (see also
306
The general rules described in the previous section can particular to the MSSM. In the case when the field content are fixed, it is more useful to deal with numerical rather Rewriting the supe~ote~tial ( 18) and the soft terms ( 19) one has
be applied to any model, in and the Yukawa interactions than with tensor couplings. in terms of group invariants,
(65)
and
where we have introduced numerical couplings ya, Mb, _&, and L?h, Usually, it is assumed that the soft terms obey the universality hypothesis, repeat the structure of a super-potential, namely
A, = YA, ,
&J = ~b~b,
Thus, we have the following
(
m2) ‘. z .I
m?@. 8
J’
set of cou~~~n~s and soft p~ameters~
i.e. they
Then, the renormalization like (for simplicity,
group p functions
we assume the diagonal
of a rigid theory (32) ?I(45)) renormalization
(47)
look
of matter superfields)
where yi is the anomalous dimension of the superfield cpi, yaj is the anomalous dimension of the gauge superfield (in some gauges) and numerical matrices K and T specify which p~ti~ular fields contribute to a given term in Eq. (65). To get the re~o~~i~atio~ of the soft terms, one has to apply the a~~o~thrn of the previous section, Eqs. (32)) (46), (48). In terms of numerical couplings it is simp~i The renormalizations in the following way:
&,y = D2yi and the operators
of the soft terms are expressed
through those of a rigid theory
9
Di and D2 now take the form
D2=
(76)
where we have used the notation
Ya - yz.
To illustrate these rules, we consider as an example one loop r~no~~ization of the MSSM couplings. Consider for simplicity the third generation Yukawa couplings only. Then, the superpotential
where Iepton and j The
is
Q, U, D”, & and E are quark doublet, up-quark, down-qu~k, lepton doublet and singlet supe~~lds, r~s~ctively, and pi1 and H2 are Higgs doublet s~pe~elds. i are the SU(2) indices. soft terms have a universal form
+A,y,qkh;
+ Ahygj=k??zj
+ A,y,l@‘h”l -I- Bph; h; + h-c.),
where the small letters denote the scalar components
(78)
of the ~o~~~~offd~n~ s~~r~~lds
and A, are the ga~g~~os. The ~~~2) indices are s~F~ressed_ ~eno~ali~ti~~s in a rigid theory in the one loop order are given by the formulae 179) (80) (81) (82) (83) (84) (85) (86)
(87) (88) (89) (90) (91)
(92) (93) (94) 195) (96) (97)
We calculate here the two and three loop gaugino mass renormalization out of a corresponding gauge p functions. The RG /3 functions for the gauge couplings in the MSSM are 1101
j
z
bqffj
hjkajak
.ik
C
-
Uif
Yf)
f
-c
aijf@jrf
$_
a;f&Y,
if
I
=+ .
‘.,
c 104)
where YJ means Y;, Yb and Y7 and the eoeffi~ie~ts b;, bij, aif, bg,?,aig and aifg are given in Ref. [ 101. For the gaugino masses we have
bijk@jak(mA;
-f= mAj
+
mAk)
-
xaijfajYf(mAi
+ mAj
-
A~IJ
if
The same fo~u~ae
can easily be obtainer
for the other soft terms. We du not write
them down here due to the lack of space, Instead, the explicit formulae in the case when only top Yukawa coupling and au3 are retained are presented in Appendix A.
Acknowledgemen D.K. is grateful to D.Maison and K.Ch~tyrki~ for valuable discussions. Fi~~~~~a~ support from RFBR grant # 9602-17379a, RFRR-DFG grant # 00082G and DFG grant # 436 RIJS 113/335/0(R) (D.K. and I.K.), and INTAS grant # 1180ext (L-A.) is girdle a~k~owl ged. D.K. and I.K. woul eir gratitude to the ~~iv~rsi~~ of K~lsrube where part of rhis work bas been done.
L.V
310
Appendix
Avdeev et al./Nucleur
A. Three-loop
Physics B 510 (1998) 289-312
renormalizations
in the MSSM
In this section, we present explicit formulae for rigid and soft term renormalizations in the MSSM in the three-loop approximation in the case when we retain only ~y3 and top Yukawa coupling
&
The rigid renormalizations f& = -3a:
+ &
14a3 - 4y,) + cr; [+q
at = (2y, - $4 +(y
are [ lo]
- (ST* + $a:)
+9653)k;*a3
- (y
- E+3Y,
+ 3Oy:])
tA.1)
+32053)~y:],
(A.21
+ [(30 + 1253)q3
+ yl3)y,cr;
+ (9
(A.3)
r,,=-~cuj--$~~:+[--~~y~+(~+320~3)cu:], YQ = (r, - +3)
- (5y: + +‘;)
+(?+48&));*a3-
+ [( 15 + 6[3)K3 (A.4)
(~+~~3)y,cu~+(~+320~3)cy~],
YH>= (3Y,) - (9Y,* - l6Yb3) +](57+
1813)K3+
(72-
144l3)?*~3
a3) - (22K2 - 16k3
Pu,=K{W-$’
- (y+
(A.5)
l64’3)Y,c~i],
+ [ (102 + 3613)I;’
+ +x;)
(A.6)
+~y,2n3-(~+288[3)Y,a:+(~+640[3)cu;]}, pjLz =,u* (3yt - (9q’ - l6xa3) + ](57+
1853)y3 + (72 - 144[3)&*a3
The corresponding
soft term renormalizations
PM3 = -3a3M3
+ 28a:M3
+347a;M3
- ya;r,(2M3
PA, = (6KAr f ~wMJ) +[(306
+ 28853)Y,&A,
- A,)
- A,) + 30a3K2(M3
+ yq2q(2Ar
- Mj) - T(Y~M~]
P,,,; =2&(r$
+ 1653)Y,&A,
- M3)] - M3) (A.lO)
- 2M3)],
+ mi -I- rni2+ A;) - ~cqM~
- 16Yr2(m: +
12i'3)&3(mf+m~ +m&+
-yQ’:M:+3(30+
(A.9)
+ 64013)a;M3],
+[( 171 + 5413)I;“At + (72 - 144&)~*q(2A, -(q
(A.8)
- 2A,),
- M3)
- 2M3) - 3( y
- [ 18y’At - 16Q3(Ar
(A.7)
+ 16[3)Y&]}.
read
- [44y*A, - l6Y,a3(A,
+ 10813)y3A,
-( y PE=~KA~
- 4@3(M3
- (y
rni
+ m&
+ 2,4:)
3Af)
+(~+965~)~2~3[(2At-M3)2+2(m~+m~+m2,2)+~~] -(~+~13)Yt~~[tA~-2M~)*+(m~+m~+m~,)+2M~] +12(? P,,,; = - +3M;
+ 320&&M;, - +r;M:
(A.1 1)
Note added
When this paper has already been finished we became aware of the paper [ 161, where similar results were obtained. Our results coincide with their ones in many points. Except for the relation between the gauge coupling /3 function and the gaugino /3 function, the authors of Ref. [It;] have deduced the same relations between the reno~alizatiu~ group functions of the soft and rigid theories through the d~~ere~t~al operators DJ and Dx starting from the Yamada’s rules [ 61. As for the &-&, relation, their starting point was the Hisano-Shifman formula [ 141. Our approach is based on a consideration of the soft theory as a rigid one embedded
into the external
x-independent
superfields,
that
are the charges and masses of the theory. The Yamada’s rules together with the operator constructions DI and Dz are just the technical consequences of this approach.
References [ 1 1 L. Girardello and M.T. Grisaru, Nucl. Phys. B 194 ( 1982) 65. [ 2 1 R. Delbourgo, Nuovo Cim. A 25 ( 1975) 646; A. Salam and J. Strathdee, Nucl. Phys. 3 86 (1975) 142; K. Fujikawa and W. Lang, Nucl. Phys. B 88 ( 1975)61. [ 3 I M.T. Grisaru, M. RoiSek and W. Siegel, Nuel. Phys. B 59 ( 1979) 429. 14 / .!.A. H~~ay~~-Net~, Phys. L&t. 6 135 { 1984) 78; F. Feruglio, J.A. ~~~a~~l-N~t~ and E Legovini, Nucf. Phys. B 249 (1985) 533; /S f M. Scholl, Z. Fhys. C 28 (1985) 545. [ 6 1 Y. Yamada, Phys. Rev, D 50 ( 19943 3537. 171 M. Vaughn and S. Martin, Phys. L&t. B 318 (1993) 331; Phys. Rev. D 50 (1994)
2282.
312
L.K Avdeev et ul./Nuclear
Physics B 510 (1998) 289-312
181 1. Jack and D.R.T. Jones, Phys. Lett. B 333 (1994) 372. [9 1 1.Jack, D.R.T. Jones and C.G. North, hep-ph/9606323, Phys. Lett. B 386 ( 1996) 138 [ IO] PM. Ferreira, 1. Jack and D.R.T. Jones, hep-ph/9605440, Phys. Lett. B 387 (1996) 80. [ I I ] P West, Introduction to Supersymmetry and Supergravity (World Scientific, Singapore, 1986). [ 121 W. Siegel, Phys. Lett. B 84 (1979) 193; R. van Damme and G. ‘t Hooft, Phys. Lett. B 150 (198.5) 133. ( 131 See, for example, V. Barger, M.S. Berger and P Ohmann, Phys. Rev. D 47 (1993) 1093; W. Boer, R. Ehret and D.I. Kazakov, Z. Phys. C 67 ( 1995) 667. [ 141 J. Hisano and M.A. Shifman, hep-ph/9705417. 1IS] 1.Jack, D.R.T. Jones, S. Martin, M. Vaughn and Y. Yamada, Phys. Rev. D 50 ( 1994) 5481. 1161 1. Jack and D.R.T. Jones, hep-ph/9709364.
Nuclear Physics B 510 (1998) 289-312
Renormalizations in softly broken SUSY gauge theories L.V. Avdeev
l,
D.I. Kazakov 2, I.N. Kondrashuk 3
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, I41 980 Dubna, Moscow Region, Russia
Received 10 October 1997; accepted 27 October 1997
Abstract The supergraph technique for calculations in supersymmetric gauge theories where supersymmetry is broken in a “soft” way (without introducing quadratic divergencies) is reviewed. By introducing an external spurion field the set of Feynman rules is formulated and explicit connections between the UV counterterms of a softly broken and rigid SUSY theories are found. It is shown that the renormalization constants of softly broken SUSY gauge theory also become external superlields depending on the spurion field. Their explicit form repeats that of the constants of a rigid theory with the redefinition of the couplings. The method allows us to reproduce all known results on the renormalization of soft couplings and masses in a softly broken theory. As an example the renormalization group functions for soft couplings and masses in the Minimal Supersymmetric Standard Model up to the three-loop level are calculated. @ 1998 Elsevier Science B.V. PACS: ll.lO.Gh; ll.lO.Hi; 11.3O.Pb Keywords: Soft supersymmetry breaking; Spurion; External superfield
1. Introduction Supersymmetry, if it exists, must be broken. The gauge theory with softly broken supersymmetry has been widely studied. To break supersymmetry without destroying the renormalization properties of SUSY theories, in particular the non-renormalization ’ E-mail avdeevl@thsunl .jinr.dubna.su. *E-mail [email protected]. 3 Email ikond@ thsun 1.jinr.dubna.su. 0550-3213/98/$19,00 @ 1998 Elsevier Science B.V. All rights reserved. PIISO550-3213(97)00706-2
theorems
and the cancellation
called soft terms
of quadratic
[ 1] , They include
divergencies,
the bilinear
one has to introduce
and trilinear
scalar couplings
mass terms for scalars and gauginos. A powerful method for studying SWSY theories which keeps s~pers~mm~try technique
[2,3].
the SQand the rna~~~est
It is also a plicable to softly hroke~ SWSY mode [ 1,4,S] AAs has been shown by Yamada [ 6 j
by using the “spuri~n~’ external s~~r~e~ds
with the help of the s~urio~ method the ca~cnlation breaking
terms
is a much
simpler
of the j3 f~~ct~on~ of soft SUSY-
task than in the component
approach.
Using
the
spurion technique he has derived very efficient rules which allow him to calculate the soft term divergencies starting from those of a rigid theory in low orders of perturbation theory. Hvwever, these rules need some modification since, as mentioned by Yamada 161, there might appear some problems in t e calcufation of vector vertices because they ~~~~~j~ cl-iii-al derivatives acting on external lines. In the present paper we develop such a modification of the spurion technique in gauge theories which takes care of the above mentioned problem. We formulate the Feynman rules and show that the ultraviolet divergent parts of the Green functions of a softly broken SUSY gauge theory are proportional to those of a rigid theory with the spurio~ fields being factorized. In genera], we consider a softly hroke~ s~~ersymmetric barge theory as a rigid SUSY theory ernb~~~ into the external space-time i~de~nde~t s~per~eld~ so that all the parameters as the couplings and masses become external superfields. This approach to a softly broken sypersymmetric theory allows us to use remarkable mathematical properties of N = 1 SUSY theories sucfi as Nan-reno~a~~zat~on theorems, cancellation of quad~t~c d~ver~e~ces~ etc. We show that the reno~a~~zat~on procedure in a softly broken SUSY gauge theory can be perfo~ed in exactly the same way as in a rigid theory with the reno~alizatju~ constants being external super~elds. They are related to the co~es~ond~~~ reno~ali~ation constants of a rigid theory by the coupling constants redefinition.
This allows us to find explicit
relations
between
the reno~ali~atio~s
of
soft and rigid couplings. Throughout the paper we assume the existence of some gauge and SUSY invariant re~u~~izatio~ and a minjma~ suhtractio~ procedure. Though it is some problem by itself, we do not consider it here. Provided the rigid theory is well defined, we consider the mod~~cations which appear due to the presence of soft SUSY breaking terms. To be more precise, in the following sections when discussing one, two and three loop calcn~at~o~s of the reno~alization constants we have in mind dimensional reduction and the minimal subtraction scheme. Though dimensional reduction is not self-consistent in general, it is safe to use it in low orders and all the actual calculations are performed in the framework of dimensional reduction [7-IO]. nevertheless, our main fo~ulae are valid within any jny~r~ant regular~zat~on, which keeps the action unchanged, and a minimal subtraction scheme.
291
2, Softly broken pun! N = 1 SUSY Yang-Mills
theory
Let us first consider pure N = 1 SUSY Yang-Mills The Lagrang~an of a rigid theory is given by
Here W” is the field strength chiral superfield
theory with a simple gauge group.
defined by
W”= o2 (e -VP%?“), where V, = VA (~A)~ and TA are the generators of the gauge group C. To perform a soft SUSY breaking, one can introduce a ga~gi~o mass term. This is the only term allowed in a pure gauge theory which does not break the gauge ~nv~i~ce. The soft SUSY breaking
term is
where /t is the gaugino field. To rewrite it in terms of superfields, let us introduce an external spurion superfield q = @*, where B is a ~r~smannian parameter. The softly broken ~grangia~ L soft=
can now be written as
s
’
d2B~(1
-2pf12)TrW”W,+/d26-${l
-2ji,t?)Tr#‘*Wa.
(2)
In terms of component fields the interaction with external spurion superfield leads to a gaugino mass equal to mu = p, while the gauge field remains massless. This external chirat superfield emerging
can be considered
from su~r~~av~ty,
as a vacuum exp~tation
value of a dilaton superfield
however, this is not relevant to further ~~ns~derat~o~.
Consider now the Feynman rules correspdnding to the Lagrangian (2). For this purpose it is useful to rewrite the integral over the chiral superspace in (2) in terms of an integral over the whole superspace.
t-l4s2 (1 Consider
(3)
> part of the action
d4x 80 d28Tr -& s
.
2p~*)Tre-“b”e”D2e-“LS,eV
the quadratic
+
One has
(1 - 2~8”) B’V( x, 8, #)D2DkV(
x, 8,8>.
To find the propagator, fixing condition
one first has to fix the gauge. In a rigid theory the gauge
is usually
D2v = f,
taken as
LFV = J;.
This is ~uivalent
(4)
to adding to the action ( I> a gauge-non-inv~i~~t
term propurtio~al
to
c gauge-fixing - -Tr By joining
dff
(ff
this together the quadratic
1t can also be rewritten 4
= -Tr &
88
+ jr).
(5)
part of the rigid action takes the form
as
d%TrV
[L-l,,? + $1
d%
s’ s where we have introduced
[ 111
the projectors
The inverse operator then gives the vector propagator (v”(e,)vqzz)).@
= ;flt~2;~?3@qZ,
The next task related
-Z2)@.
to the determination
of the vector propagator
asso&~ated ghost term. Under the gauge tra~sfo~atio~
is to find the
the sup~~~e~d V is transfo~ed
as
or in the infinitesimal
form
OL
v+V+H(V)R+R(V)A=V+ ~o~se~ue~tly,
c ghost =
the ~so~~ated
s
=
-p
(H(V)c+A(V)E~
d28d28
(b+t;)
where b and c For a system in order to take The reason for
Tr;
(A + A) + coth
(g)A (n-n)].
ghost term is
d2@Tr$bD2
s
A
-f O[
+/d28Tr$6L?’ A (c+C)
(N(V)c+l?(V)L;)
+coth
are the ~add~v-Popov ghost chit-al superfields. in the external spurion field we can change the gauge fixing condition (4) into account the interaction of the associated ghost fields with the spurion. this is the appearance of these terms in due course of renormalization.
Here, we depart from the Feynman
rules of Ref. [ 61. Our aim is also to get the same
common spurion factor for the ghost propagator as for the vector one, as it will be clear below. For these purposes we choose the gauge-fixing condition in the form
or
where we have introduced
the notation
22 = g2 (I + #iLiP+ /.&a2+ ~~=~~~~) . Note that due to the ~rassma~nia~
Now the gauge-fixing
origin of B
term can be written in the standard form
t be a real ~-independent
which leads to the following
quadratic
s~pe~~~~d. l[n what follows we take
part of the action in a softly broken theory
One now has to find the inverse operator, It can be done analogously
to the rigid case
by introducing the projection operators. One should take into account, however, that while integrating the ~o~a~a~t derivatives by part they may also act on the sp~rjon field. Strictly
speaking,
this leads to additional
terms in the propagator
which are, however,
suppressed by powers of momenta and are iness~ntial in the analysts of UV diverge~~es~ To understand it better, one can treat the spurion terms in E&q_C101 as inte~a~tio~s~ and to find the propagator, one has to take into account an infinite chain of this kind of insertions adding up into a geometrical progression. It can be done by the method of Ref. [ 5 1. If in due course of this procedure some of the covariant derivatives act on a spurion field, one does not get enough powers of momenta to cancel the denumi~ato~ and the resulting terms are thus suppresse for a high momentum square. Thus, as it has been also noted in Ref. [6], for the calculation of the divergent parts of the Green functions it is suf~~~e~t to take the part of the vector propagator where the chiral derivatives do not act on spurions.
Having this in mind, one can proceed in full analogy with tbe rigid theory, replacing the coupling
g* by g2, and get
where by i~eleva~t
terms we mean the ones d~reasing
faster than l/p2
Thus, we get a simple relation between the vector propagators
+irrelevant
for large y2.
of soft and rigid theories:
terms.
Notice that control to Ref. 161 this relation is valid for any choice of Q. The change of the gauge fixing condition leads to the change of the ghost Lagrangian
(6). It becomes
As one can see from Eq. ( 12), the situation with the ghost propagator that contain ghost superfields derivatives in the ghost-vector
and the vertices
is more simple. This is due to the absence of chiral interaction. Nevertheless, when calculating the inverse
operator one has to repeat the same proc~ure with the cov~ia~t derivatives arising from the source terms. In complete analogy with the case of the vector propagator the essential derivatives
part of the ghost propagator do not act on the spurion
powers of momenta. (G(z~)~(z~))~~~~
is obtained
by ass~mjng
that the covariant
field. The other terms are suppressed
by the
I-Ience, one again has = @‘/g’)
(C(zj
)G(II))~~~~~
+
irrelevant
terms,
(13)
where G stands for any ghost superfield. Thus, to perform the analysis of the divergent part of the diagrams in a soft theory, one has to use the same propagators as in a rigid theory multiplied by the factor g*/‘g’. It is also obviously true for any vertex of the ghost-vector interactions of the softly broken theory ( 12). Each vertex of this type has to be multiplied by the inverse factor g2/g? The situation is less obvious with the vector vertices. Here, one has two terms as in Pq. (2) which differ only in the order of cov~iant derivatives, one being complex conjugates to the other. Per~o~ing ~rassmannia~ algebra in any diagram one can always replace one term by the other; so actually one has only one term. This is what we have in a rigid theory. In a soft theory the situation is similar but the first term is
multiplied by a factor (1 - 2pe2) , while the second by the complex conjugated one (1 - 2~6~). Hence, one can consider the vector vertices of only one type, as in a rigid theory, multiplied by a factor (1 - ~8~ - pe2), i.e. by g2/@ as in the case of the ghost-v~tor vertices. Thus, we see that any element of the Feynm~n rules fur a softly broken theory the ~o~es~ondi~g element of a rigid theory factor which is a ~olynomi~ in the Grossman ~oord~nates~
2.2. The ultraviolet
counretierms
Consider now the Green functions of a softiy broken theory ~~nstru~t~ with the help of the Feynman rules delved in the previous section. Due to the presence of a factor g2fg” in the vertices and of the inverse one in the propagators, any ~eynma~ d~a~rarn in a broken theory is given y that of a rigid theory with the spurion factors on the lines and vertices. Our aim is to show that these polynomials in grassmann variables can freely flow through the diagram and are finally collected in front of a diagram as a common factor. Indeed, consider t ~o~es~o~di~g ~rassrna~n~~ ~~tegraiio~~The usual pro~~~~e of evaluation of Grassma~n~~ integrals in a rigid theory is the following: 63 If there are no chiral derivatives II, or ija on the line, it is propo~ona~ to the ~-function and can be shrunk to a point. (ii) If there is some number of derivatives, one can remove them from a line integrating by parts. (iii) Repeating this procedure several times and using the prosodies of chiral derivatives (D,, Da) = {&, II,) = 0,
D,D@ = r,@,
D2B”D2 = 16pW,
etc.
to reduce their number one can finally come to a final integration where all the chiral derivatives are concentrated on one line. The powers of momenta which appear in due course af this procedure cancel some propagator lines so that the resulting diagram diverges. The final integration is ~rfo~ed with the help of the relation
Consider what happens if one adds Grassman~i~n polynomials en the lines and vertices. Suppose first that all chiral derivatives inside a graph are contracted and integrated inside the graph so that there are no external derivatives left. If one of these internal derivatives acts on a s~urio~ field it creates additional 8” instep of II”,. Then either n survives till the last integration, or some other D, acts on it. Thus, at the last step of integration one has instead of ( 14) either
It is easy to see that both the expressions
are zero. Thus, the configuration
chiral derivative acts on a spurion field gives no contribution.
field flows through the diagram freely and can be factorized. Consider now the case when a graph contains some derivatives lines when perfo~in~
the Grassmannia~
when the
This means that the spurion that act on external
i~te~rat~on~ This case ~o~esponds
to vector
interactions. Let some covariant
derivative,
which originally
was inside the diagram,
pass through
the diagram during the above mentioned procedure and act on the external vector line. We can trace this derivative in the diagram of the soft theory. On its way through the diagram
it can meet some “spurion”
factor in a vertex or a propagator.
Integrating
by
parts one has two terms. The first one, when the chiral derivative acts on a spurion, and the second one, when it acts on the nei~~bouri~g propagator. When the ~ov~ja~t derivative does not act on a sp~r~on it goes through the diagram like in a rigid theory and acts on external line. As a result one gets the expression g2 fDLYVMz). In the other case, when the covariant derivative acts on the spurion, the number of covariant derivatives is reduced compared to the rigid theory. Then, either one does not have enough of them to cancel the powers of momenta in the denominator or to get the non-zero answer or one has the same diagram as in the rigid case, but with the factor Dg*, that goes outside the diagram. Independently of the position of this factor inside the diagram, the nal expression one gets after the ~al~~~atio~ of the diagram contains ~~*~) V’“’which together with the usual contribution, gives II, (j2V’“) instead of g”D,V”’ in the rigid case. All together these factors S2, which are met by the chiral derivative on its way to the external line of the diagram, are collected in a monomial R(S2), Therefore, the final expression for the diagram contains D, (R(S2) V”). The same conclusion is true for all derivatives of t e divergent vector dia~r~nl that act on the external lines. Since the cou~terterms in a given order of perturbation theory should be gauge invariant, in the final expression only the countertetms with four covariant derivatives survive. The have the form
where Rr , R2, R3 and F&tare some monomials in $. This expression witk the ~o~es~on~~~~ ~ounterte~ of the rigid theory
should be compared
R(g2)V~rD”V”L52VkD,V’, where
The momentum integral for the diagram in a soft theory is identical to &at in a rigid one since the algebraic operations with the covariant derivatives coincide for the both cases.
We now want to argue that the monomials
Ri(g2) can stand anywhere in Eq. (IS),
i.e.
that the spurions can flow through the diagram. Indeed, in a rigid theory the counterterms are absorbed
into the redefinition
of the field V and the coupling
g2
vs = z~~2(g2~v) & = qg2>g2* In a soft theory Ri are the functions malization
constants
become
ctf the external
these functions
sp~rion s~~~~el~
too. Collecting
type (15)) one should get an expression which being quantities obeys the gauge invariance, namely
and the renor-
alt counterterms
written
of the
in terms of the bare
where the primed rno~orn~~s mean that pat-Q they are absorbed into the ~e~o~aIization of the field Vs. Now it is clear that the covariant
derivatives
in Eq. (16)
do not act on monomials
Ri(g2>. Indeed, if one of the covariant derivatives D, in Eq. ( 16) acts on the monomials Ri or R& the two exponentials cancel each other and one is left with Tr W” that is equal to zero. In the case when both the chiral derivatives in Eq. (16> act on Ri and Ri in both cases the expo~e~tials cancel and one has a divergent constant that should be absorbed into the reno~ali~tio~ of the vacuu density. Hence, one can take the monomials R$ and Ri out of the cov~ia~t derivatives D,. The other derivative, B2, does not act on the monomial for the reason that e-P4D”eG*82 = 0 so that in the bracket in Eq. ( 16) only the chiral spurion q = O2 survives and fi2 does not act on it. Thus, we come to a conclusion that the monomials can be factorised in front of the expression
in Eq. ( 16) just like in the rigid theory and the only difference
rigid and softly broken theories is that the coupling gz9 i.e.
constant
2
between
the
should be replaced by
Zi = Zj (g* + g”] *
3. Softly broken SUSY gauge theory with chiral matter Consider
now a rigid SIJSY gauge theory with chiral matter. The ~agrangian
in terms of s~pe~elds Lrigid
d28d28&‘(eV)i@j
=
written
looks like + /dZOV+~c.i26+i.J,
(17)
s
where the superpotential
W in a general form is4
W = ~~~~k~~~j~~ + ~~ij~~~j.
(181
4 To avoid further c~rn~l~c~~~~s (see, e.g. Ref. 161) we do not introduce linear terms in Eqs. ( 18), ( 19) s This corresponds to the case when none of the fields goes into vacuum. In the absence of singlets this is guaranteed by the gauge invariance imposing some restrictions on the couplings.
The SUSY breaking terms which satisfy the requirement
of “‘softness” can be written as
Like in the case of a pure gauge theory the soft terms (19) can be written down in terms of superfields
by usin
the external
spurion
field. Thus, the full Lagrangian
for
the softly broken theory can be written as
The ~agra~gia~
(~~~ arrows one to write down the ~ey~rnan rules fur the matted field
propagators and vertices in a soft theory. We start with the ~r~pa~at~r of the chiral field. For the purpose of the analysis of diver~~~~es ane can jgn~re the mass terms M’s and Bli, since in the minimal scheme they do not ~~~trib~te to the UV divergences= Then, the quadratic part looks Iike
The inverse operator is easy to obtain due to the nilp0tent
~ha~a~te~ of the sprain
fields.
One easily gets
$-irrelevant
terms,
ere tbe ~~~le~a~t terms again arise f~~rn c~variant derivatives
acting on spurions and
am suppressed by ~~~~~s of momenta. The vender-mattes vertices, a~~~rdi~g to Eq, (ZO), gain the factor (8: - {~~~~~~~ so that if in a dia~r~ one has an equal n~~~~r of chiral ~ro~a~at~rs and vector-u~att~r ~~rti~~s the spurion factors cancel. The chiral vertices of a soft theory, as it follows from Eq. (XI), are the same as in a rigid theory with the Yukawa couplings being replaced by i$, _ xi8 _ ~~jk~ f A_
;iij& + ;iijk - ~i~~~ .
The structure of the W counterterms in cbiral vertices is sirn~l~ to that of the vector vertices, but is simpler due to the absence of the covariant derivatives on external lines. This corresponds to the first case considered in the previous section. To get the UV divergent diagram, the ~ovar~ant derivatives should nut act on $~~r~~n fields, which
L.V Avdeev et al./Nuclear
means that sp~rions
factorize.
with the ehiral propagators, couplings
Physics 3 510 (1998) 289-312
In the diagrams,
299
when the chiral vertices are ~on~ra~t~
this results in the fo~~uwing effective ehange of the Yukawa
of a softly broken theory
Thus, the WV ~o~~terte~s
of a softly broken theory are obtained
from those of a
rigid one by the substitution
4. Renormalization of soft versus rigid theory: the general case The external
ma~izatio~
field ~o~st~~t~o~ ~es~r~~
of soft terms starting
above allows one to write down the renor-
from the known
without any new diagram calculation.
The following
reno~~~zat~on statement
of a rigid theory
is valid:
The statement Let a rigid theory (I), (18) be renormalized via introduction of the renormalization constants Zi, defined within same minimal subtraction massless scheme. Then, a softly broken theory (21, (20) is renormalized via introduction of the renormalization s~~e~e~ds 2~ whack are related to Zi by the ca~p~i~g ca~sta~ts r~d~~~it~a~
where the redejned couplings are
This allows us to find explicit relations between the renormalizations couplings
which is an explicit realization
of soft and rigid
on the level of final expressions
of the rules
A and I3 of Ref. [6]. The re~o~a~izatjon constants for vector s~~er~elds and gauge couplings are general s~per~el~s, i.e. ey depend on 71 and ji, white those for the chiral matter and ghost fields and the greeters of a supe~ote~t~~ are chiral supe~elds, i.e, they depend either on 71 or *. From Eqs. (24) and (25)-(27) it is possible to write down an explicit differential operator which has to be applied to the p functions of a rigid theory in order to get those for the soft terms, We first construct this operator in a general case and compare the resulting expressions with explicit calculations made up to two loops. Then, using
the formulated
algorithm
we calculate some three-loop
soft term p functions.
section, we consider some particmar models. To simplify the formulas hereafter we use the following
Consider
first the gauge couplings
In the last
notation:
ari. One has
wkere Zai is the product of the wave function
and vertex re~o~a~~zation
constants.
Though & and Zaj are general s~~er~e~ds, one has to consider only t since the Lagrangian ( 1) consists of two terms, a chiral and an antich~~al, and in each term only the proper chirality part contributes. Therefore, we consider the chiral part of Eq. (28)
aiding
over q one has
where the operator 131 ~xt~~c~s the hnear with respect Eqs. (25)-(27) the explicit form of Dr is
Combining
to 71 part of Z,i(Ly).
Due to
Eqs, (29) and (30) one gets
To find tke co~espo~d~n~ p functions one has to differentiate Eqs. (29) and (31) with respect to the scale factor bav~n~ in mini tbat the operator Dr is scale invariant~ This gives
ere yLyi is the logaritbm~c derivative of In Z&i equal to the a~orna~o~s dimension of the vector superfield in some particular gauges. This result is in complete co~es~ondence with that of Ref. [ 141, Indeed, from Eqs. (32) one can derive that the ratio ~~~/~~ is reno~al~zation group invariant.
301
4.2. Chiral matter Consider
now the chiral matter. Due to the non-renormalization
reno~alization
of a supe~orential.
the wave functions.
theorems, there is no
This means that the only reno~alization
The ~o~es~o~diag
term in the Lagr~gi~
comes from
in~l~d~~g reno~alizations
looks like
s
d20 $6 @g.j@ 1 .it
where the renormalization
(33) superfield
2; now has a decomposition
2; = &+& (6; + A~~~) z&,
(341
where the chiral( ant~chira~) reno~a~~zatio~ renormalizations
super-fields Zij ( Zi+k ) are the wave function
and .4f is the soft term (m2)j
Zj( &, jr, F) over the grassmann
renormalization.
To find, them expand
variables
-;f;;= 2; + Dt z$j + D, z;?j + DzZj~?j,
(35)
where the operator I)1 now has to take into account the q de~n~e~ce of the Yukawa ~ou~~j~g y “jk, D 1 is conjug ated to Dr and I)2 is a second order ~i~ere~t~al o~e~tor ich extracts the q+j dependence of 2;. They have the form D1 =mAiai-
a &2!j
+yabc-
d
ahbc
Substituting
+YbUC-
-
a
a
D] = mAiai_ - Aijkda; @ijk ’
(361
a
(371
@bnc
this into PEq. (34) one has
~~,i=Z~j+ i~j=Z~j
a
- A ijk dy”jk’
(Z$‘)~D,Zjk71=Z~k(Sjkf(Z-‘)f’D,Z/!rl), + B,Z~(Z~‘)jkd=
~~=(Z~~~~D~~~(Z~‘)~
CS~+ ii,Z~(Z-‘)fii)Z~j
(381 ))
(39)
- (Z~‘)lt>lZ~(Z-“>j;D~Z~(Z~‘)J”,
C4Q)
where Z~j satisfies Z~k Z~j = Z~ )
and is the square rout of Zi in the perturbative sense. The inverse one is (;?;‘>; = (8; - {Z-‘~~D~Z~~~~Z~‘~~. We can now write down the relations between the renormaliz of a superpotential. One has
(411
the bare couplings
(42) (43)
e last line can also
differentiating
wit
respect to a scale factor one gets after some alge
with L$ given by Eq, (40). ~iffer~~t~~t~n~ it with respect to a scde factor and using the explicit form of the seeon order d~ffc~ntja~ operator &, after some algebra one obtains a simple ~xp~~ssio~ for the /I function
Relations between the rigid and soft terms ~en~rrnali~t~~ns snbtractio~ scheme are surnrn~~~ as foli~ws;
in a m~ssless
minimal
L.I! Avdeev et al./Nuclear
The rigid terms
The soft terms
Pa, =
Pm,,
aiYa,
pjc, = L ( Milyi 2
1
+ j@#)
p;
1
= D
1Yni
= ; (&yli
+
f+jyj)
-(M’*D,d pijk ?
=
1( #jlyk 2
1
+
#lkyi
+
1
yijkyi)
px”
1
= 4 ( Aijryf
+ M’jDly()
+
-(yij’Dly; (Pm2>j =
a
- A ijk ayijk’ JCZi
&=DiO
+mi,ai& +$(m2)~(ynbc--& I d
a ‘%‘nbc
Ailk+
I
+
A’jkyi
1
>
+ yiikD,j I + y’jkD,yi)1
D2r;
d
D1 =mA;ai-
+hbc
303
Physics B 510 (1998) 289-312
+
Ybuc &‘bnc
+
d ybcn aYbcn
+ybnc--&+ybcn---&
>
5. Illustration To make the above formulae more clear and to demonstrate how they work in practice, we consider the renormalization group functions in a general theory up to two loops. We follow the notation of Ref. [8] except that our p functions are half of those of Ref. [8]. Note that all the calculations in Ref. [8] are performed in the framework of dimensional reduction and the MS scheme. The gauge ,6 functions and the anomalous dimensions of matter superfields in a massless scheme are the functions of dimensionless gauge and Yukawa couplings of a rigid theory. 5. I. One-loop renormalization In the one-loop order, the renormalization group functions simplicity, we consider the case of a single gauge coupling) y;‘) =cYQ, 7; (‘)=
of a rigid theory are (for
Q=T(R)-3C(G),
(51)
$y'k'yjkl -2&(R);,
where the Casimir operators T(W~AB=T~(RARB),
(52) are defined by C(G)~A~I
Using Eqs. (32) -( 50)) we construct terms
= ~ACD~B~D,
the renormalization
C(R);
= (R,~RA)~.
group functions
for the soft
L.c( Avdeev et al./Nuclear
304
Physics B 510 (1998) 289-312
pm ntn = a%@,
Pi (1) = $“(
(53) ;y+nrylk,, - 2aC( R);)
+M”( kAikn’yjknl + 2amAC( R){) + (i t-f j) , ijk (I)
PA
=
(54)
;Ai”( ;ykn”‘yln,,, - 2&(R);) +y’j’( $Ak”‘“yj,,,,, + 2amAC ( R) :) +(i cf j) + (i *
k) ,
(55)
- 4amf,C(R)j
[&2]1 (I) = iAik’Ajkl
f~ynk’(m2)~y,;kl
f iyik’(m2)gy&
+ iy’S’(m2):y,jk[.
(56)
One can easily see that the resulting formulae coincide with those of Ref. [ 81 with one exception: we have ignored hereafter all the tadpole terms assuming that they all are equal to zero because of the absence of linear terms in the action (see footnote 4). 5.2. Two-loop
renormalization
In two loops the situation y’*’ a =2a*C(G)Q y.Y2)= -(fn”‘yjnrr~
is more tricky. The rigid renormalizations
+ 2aC(R),y&)
Then, the soft renormalizations
r = dim G = aA~ , (57)
&@‘y;k[ - 2&(R)!),
- +(R);(
are
+ ~cz*QC(R)~.
( $y”k’y,>k[- 2aC(R)i)
are as follows:
2amA p,jt) = 4a*mAC (G) Q - -C(R)j( r
;yjk’yjkl - 2cuC(R)I’)
+~C(R);(IA’X’y~k,+2cumnC(R):), p;
12) = +“(yjkyy[kn
(59)
+2aC(R):‘Gj;)(;y”“‘y,,,,
-Mi’(AjkPylk,
- 2cxC(R);)
- 2am~C(R)I;@,)(~y”“‘y,,.s,
-kf”(yikpy[k17 +2aC(R):‘6~)(~A”“‘y,,,~, +B”a*QC(
+ 2cuC( R):‘$)
+AlJ’a2QC (R):
++Y””
j) + (in
Atk”A,jkn
+
(m”)
;Yjm
.$
(60) R);)
- 4y’*‘cx*QC (R):mA
-yij’(yknl”ylnln + 2aC( R)y#)
(*I = -(
+2amAC(R)i)
( +y”Sfyl,,,, - 2d(
-yij’ ( Ak”‘pyl,l,,7- 2cum,&(R):‘S:)(
[&I;
-2&(R);)
R){ - 4Mi’cu2QC( R)im,., + (i tf j),
Pi? (*I = -;Aii’(yk”‘f’yl,,,,
+(iw
(58)
;y”S’y,,,Yt- 2cuC(R);)
( iA”S’y[,,Yt+ 2amAC( R);)
k), (m2)&‘kpyjkn
+
+Yik”
(61) +
iyik”ylk,,
(m2)fyJk,, +
(m*)j
iyikp
(m*
),“lY.jk~
305
C~mp~ing these formulae with those of Ref. 181 we find that they coincide with one ization of m2 In Eq. (16) uf Ref. [8] has two extra terms, exception. The resow ~~4~~~ “,5 and the term ~ropo~ion~ to the mass of the s~-c~~ed e-scalars, $8. The first term, though it is not ~roport~o~~ to e2, still has tbe same origin. It has appear as a result of the s-scalar mass co~nte~erm in one loop. The presence or absence of these terms is ~eno~a~~~tio~
scheme dependents
The
version of dimensional reduction adopted in Ref. [ 81 ~o~es~o~~s to d~agr~ by diagram minimal subtractive of divergences and naturally incI~des the ~-scalar mass ~ou~ter~e~~
being formulated in the su~er~eld fo~alism no surprise that the co~es~o~di~g terms do not appear.
Our substitution
does not contain e-scalars;
There are two ways to resolve the noticed dis~r~p~~y. Either to r~e~ne the rebornovation scheme in such a way that these terms are not present, like the one in Ref. [ 121 I or to rn~~fy the dis~~sson
the ~uper~eld re~o~a~ization of this point in Ref. [ 151).
procedure to reproduce these terms (see also
306
The general rules described in the previous section can particular to the MSSM. In the case when the field content are fixed, it is more useful to deal with numerical rather Rewriting the supe~ote~tial ( 18) and the soft terms ( 19) one has
be applied to any model, in and the Yukawa interactions than with tensor couplings. in terms of group invariants,
(65)
and
where we have introduced numerical couplings ya, Mb, _&, and L?h, Usually, it is assumed that the soft terms obey the universality hypothesis, repeat the structure of a super-potential, namely
A, = YA, ,
&J = ~b~b,
Thus, we have the following
(
m2) ‘. z .I
m?@. 8
J’
set of cou~~~n~s and soft p~ameters~
i.e. they
Then, the renormalization like (for simplicity,
group p functions
we assume the diagonal
of a rigid theory (32) ?I(45)) renormalization
(47)
look
of matter superfields)
where yi is the anomalous dimension of the superfield cpi, yaj is the anomalous dimension of the gauge superfield (in some gauges) and numerical matrices K and T specify which p~ti~ular fields contribute to a given term in Eq. (65). To get the re~o~~i~atio~ of the soft terms, one has to apply the a~~o~thrn of the previous section, Eqs. (32)) (46), (48). In terms of numerical couplings it is simp~i The renormalizations in the following way:
&,y = D2yi and the operators
of the soft terms are expressed
through those of a rigid theory
9
Di and D2 now take the form
D2=
(76)
where we have used the notation
Ya - yz.
To illustrate these rules, we consider as an example one loop r~no~~ization of the MSSM couplings. Consider for simplicity the third generation Yukawa couplings only. Then, the superpotential
where Iepton and j The
is
Q, U, D”, & and E are quark doublet, up-quark, down-qu~k, lepton doublet and singlet supe~~lds, r~s~ctively, and pi1 and H2 are Higgs doublet s~pe~elds. i are the SU(2) indices. soft terms have a universal form
+A,y,qkh;
+ Ahygj=k??zj
+ A,y,l@‘h”l -I- Bph; h; + h-c.),
where the small letters denote the scalar components
(78)
of the ~o~~~~offd~n~ s~~r~~lds
and A, are the ga~g~~os. The ~~~2) indices are s~F~ressed_ ~eno~ali~ti~~s in a rigid theory in the one loop order are given by the formulae 179) (80) (81) (82) (83) (84) (85) (86)
(87) (88) (89) (90) (91)
(92) (93) (94) 195) (96) (97)
We calculate here the two and three loop gaugino mass renormalization out of a corresponding gauge p functions. The RG /3 functions for the gauge couplings in the MSSM are 1101
j
z
bqffj
hjkajak
.ik
C
-
Uif
Yf)
f
-c
aijf@jrf
$_
a;f&Y,
if
I
=+ .
‘.,
c 104)
where YJ means Y;, Yb and Y7 and the eoeffi~ie~ts b;, bij, aif, bg,?,aig and aifg are given in Ref. [ 101. For the gaugino masses we have
bijk@jak(mA;
-f= mAj
+
mAk)
-
xaijfajYf(mAi
+ mAj
-
A~IJ
if
The same fo~u~ae
can easily be obtainer
for the other soft terms. We du not write
them down here due to the lack of space, Instead, the explicit formulae in the case when only top Yukawa coupling and au3 are retained are presented in Appendix A.
Acknowledgemen D.K. is grateful to D.Maison and K.Ch~tyrki~ for valuable discussions. Fi~~~~~a~ support from RFBR grant # 9602-17379a, RFRR-DFG grant # 00082G and DFG grant # 436 RIJS 113/335/0(R) (D.K. and I.K.), and INTAS grant # 1180ext (L-A.) is girdle a~k~owl ged. D.K. and I.K. woul eir gratitude to the ~~iv~rsi~~ of K~lsrube where part of rhis work bas been done.
L.V
310
Appendix
Avdeev et al./Nucleur
A. Three-loop
Physics B 510 (1998) 289-312
renormalizations
in the MSSM
In this section, we present explicit formulae for rigid and soft term renormalizations in the MSSM in the three-loop approximation in the case when we retain only ~y3 and top Yukawa coupling
&
The rigid renormalizations f& = -3a:
+ &
14a3 - 4y,) + cr; [+q
at = (2y, - $4 +(y
are [ lo]
- (ST* + $a:)
+9653)k;*a3
- (y
- E+3Y,
+ 3Oy:])
tA.1)
+32053)~y:],
(A.21
+ [(30 + 1253)q3
+ yl3)y,cr;
+ (9
(A.3)
r,,=-~cuj--$~~:+[--~~y~+(~+320~3)cu:], YQ = (r, - +3)
- (5y: + +‘;)
+(?+48&));*a3-
+ [( 15 + 6[3)K3 (A.4)
(~+~~3)y,cu~+(~+320~3)cy~],
YH>= (3Y,) - (9Y,* - l6Yb3) +](57+
1813)K3+
(72-
144l3)?*~3
a3) - (22K2 - 16k3
Pu,=K{W-$’
- (y+
(A.5)
l64’3)Y,c~i],
+ [ (102 + 3613)I;’
+ +x;)
(A.6)
+~y,2n3-(~+288[3)Y,a:+(~+640[3)cu;]}, pjLz =,u* (3yt - (9q’ - l6xa3) + ](57+
1853)y3 + (72 - 144[3)&*a3
The corresponding
soft term renormalizations
PM3 = -3a3M3
+ 28a:M3
+347a;M3
- ya;r,(2M3
PA, = (6KAr f ~wMJ) +[(306
+ 28853)Y,&A,
- A,)
- A,) + 30a3K2(M3
+ yq2q(2Ar
- Mj) - T(Y~M~]
P,,,; =2&(r$
+ 1653)Y,&A,
- M3)] - M3) (A.lO)
- 2M3)],
+ mi -I- rni2+ A;) - ~cqM~
- 16Yr2(m: +
12i'3)&3(mf+m~ +m&+
-yQ’:M:+3(30+
(A.9)
+ 64013)a;M3],
+[( 171 + 5413)I;“At + (72 - 144&)~*q(2A, -(q
(A.8)
- 2A,),
- M3)
- 2M3) - 3( y
- [ 18y’At - 16Q3(Ar
(A.7)
+ 16[3)Y&]}.
read
- [44y*A, - l6Y,a3(A,
+ 10813)y3A,
-( y PE=~KA~
- 4@3(M3
- (y
rni
+ m&
+ 2,4:)
3Af)
+(~+965~)~2~3[(2At-M3)2+2(m~+m~+m2,2)+~~] -(~+~13)Yt~~[tA~-2M~)*+(m~+m~+m~,)+2M~] +12(? P,,,; = - +3M;
+ 320&&M;, - +r;M:
(A.1 1)
Note added
When this paper has already been finished we became aware of the paper [ 161, where similar results were obtained. Our results coincide with their ones in many points. Except for the relation between the gauge coupling /3 function and the gaugino /3 function, the authors of Ref. [It;] have deduced the same relations between the reno~alizatiu~ group functions of the soft and rigid theories through the d~~ere~t~al operators DJ and Dx starting from the Yamada’s rules [ 61. As for the &-&, relation, their starting point was the Hisano-Shifman formula [ 141. Our approach is based on a consideration of the soft theory as a rigid one embedded
into the external
x-independent
superfields,
that
are the charges and masses of the theory. The Yamada’s rules together with the operator constructions DI and Dz are just the technical consequences of this approach.
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