- Email: [email protected]
Gauge mediation without a messenger sector
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
Gauge Mediation Without A Messenger Sector Hitoshi Murayamaab* aDepartment of Physics, University of California Berkeley, CA 94720 bTheoretical Physics Group, Lawrence Berkeley National Laboratory University of California, Berkeley, CA 94720 I present a model of gauge mediation without a messenger sector. This is the first phenomenologically viable model of this type. The sector of dynamical supersymmetry breaking couples directly to the standard model gauge groups. The apparent gauge unification is preserved despite the direct coupling. The inverted hierarchy mechanism plays a crucial role in dynamics. The model is completely chiral, and hence there is no need to forbid mass terms by hand. Still vector-like messengers emerge as a consequence of dynamical breakdown of gauge symmetries.
1. I n t r o d u c t i o n
The dynamical supersymmetry breaking (DSB) [1] is an attractive idea to explain the hierarchy between the electroweak scale (or m z = 91.2 GeV) and the fundamental scale of nature M, possibly the GUT-scale (2 x 1016 GeV), the string scale (5 x 1017 GeV) or the reduced Planck scale (2 × 1018 GeV). The electroweak scale is protected by the supersymmetry up to all orders in perturbation theory but is generated by nonperturbative effects through dimensional transmutation, m z ", M e x p ( 8 7 r 2 / g 2 ( M ) b o ) << M with an asymptotically free gauge theory bo < 0. However, the implementation of this idea has been impeded by the technical difficulties of model building and the lack of understanding in non-perturbative dynamics of supersymmetric gauge theories. Since the latter problem has been largely resolved by recent works of Seiberg and his collaborators (for a review, see e.g., [2]), it is a pressing question whether one can build realistic models. *This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC0376SF00098 and in part by the National Science Foundation under grant PHY-90-21139, and also by Alfred P. Sloan Foundation. 0920-5632/98/$19.00 1998 Elsevier Science B.V. PII S0920-5632(97)00670-1
It was the pioneering works by Dine and Nelson [3] (later together with Shirman and Nir [4]) which demonstrated that one can implement the DSB in a phenomenologically viable manner, by means of gauge mediation (GM) [5]. The models presented in these works had two major important features: the hierarchy explained by the DSB and an automatic squark degeneracy thanks to the GM. Unfortunately the structure of the models was quite complicated. There are three sectors in the models which are "insulated" from each other: the DSB sector, the "messenger sector," and the supersymmetric standard model (SSM); see Fig. 1. The need for the DSB and SSM sectors is obvious; what bothered me is the apparent lack of motivation for the messenger sector. There are a couple of awkward features in the messenger sector: (1) It has to be put in by hand for no other reason. (2) It has to be "insulated" from the other two sectors in the superpotential, with only interactions being the messenger U(1) to the DSB sector and the SSM gauge interactions. (3) The "messenger" fields are vector-like under the SSM gauge interactions while their mass terms have to be forbidden. (4) Fields which are completely gauge singlets play major roles in the GM. The claim in this talk is that one can eliminate the messenger sector entirely (see [6] for other attempts to simplify the GM models). The DSB
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
290
IO,O00TeV
DSB
~
lOOTeV
1 TeV
100 GeV
DSB
Sector
? TeV
Sector
mes$
IMessenger Sector
1 TeV 100 GeV
SSM Sector
Figure 1. Structure of the original models of gauge mediation. The label GSM stands for the standard model gauge groups, and U(1)mess for the messenger U(1) gauge group. The energy scales are only approximate.
sector and SSM sector are directly coupled to each other (see Fig. 2) and the model I present is the first phenomenologically viable model of this type. It moreover has several nice features: (1) The model is completely chiral and there is no need to drop mass terms by hand. (2) There are no gange-singlet fields. (3) The apparent perturbative unification of gauge coupling constants is preserved. 2. P r o b l e m s w i t h D i r e c t G a u g e M e d i a t i o n
In the past year, there has been an intense activity to build models of GM in which the DSB sector and SSM sector are directly coupled via SSM gauge interactions and hence there is no need for a separate messenger sector (see Ann Nelson in this proceedings [7]). Of course, this is probably the most natural way to build the GM models and was indeed discussed already a decade ago. However, there used to be a major difficulty with this idea. If two sectors are directly coupled through the SSM gauge interactions, the DSB sector must have a large (un: broken) global symmetry which can accommo-
SSM Sector
Figure 2. Structure of the gauge mediation model without a messenger sector. The energy scale of DSB is model-dependent.
date the full SSM gauge group (hopefully even the GUT-gauge group such as SU(5) or SO(10)). The smallest DSB model which was known to accommodate SU(5) global symmetry back then has the dynamical gauge group SU(15), which results in 15 additional fields in 5 representation of SU(5) and lets all SSM gauge couplings to reach their Landau poles below 107 GeV. Even though there is nothing a priori wrong with such a behavior of gauge coupling constants, especially in view of the duality among the supersymmetry gauge theories, it is disheartening to throw away the apparent gauge unification which hints towards supersymmetric GUT or string unification. This problem can be solved if the DSB sector has several mass scales. The essential ingredient in this approach is that there is a field S which generates masses for the (effective) messengers Q, Q through the superpotential coupling W = XSQQ with very different sizes for the expectation values for A- and F-components, (S) 2 >> (Fs). The size of supersymmetry breaking effects in the SSM is characterized by the ratio ( F s ) / ( S ) and it must be kept around 104 GeV. In the original attempt of the direct GM, they are comparable in magnitudes: (S) ,~ (Fs) 1/2 ,,~ 104 GeV, and one finds that there are many messengers (15 of them) at 104 GeV which make all the SSM gauge coupling constants blow up quickly. On the other hand, if (S) >> (Fs) 1/2, the messenger fields can be heavy enough not to
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
0.8 gnon_pert
r.~
i
0.6
~'~ 0.4
'\
0
i
20
40
S
60
80
100
Figure 3. The scalar potential in models with a flat direction lifted only by a non-renormalizable operator. Both axes have arbitrary scales and are labeled only for the sake of emphasis that the horizontal scale is linear.
make the gauge coupling constants blow up while keeping the size of scalar, gaugino masses in the SSM sector at the electroweak scale. For instance, if (Fs) ~ (10 9 GeV) 2 and (S) ~ 1014 GeV, there is basically no concern about the Landau poles while the size of the superparticle masses comes out correctly. The very question then is how to generate the hierarchy (S) >> (Fs) 1/2. There are at least two logical possibilities. One is to employ a model which already has a mass scale built in, such as non-renormalizable models with Planck-scale suppressed operators. The other is to generate a large (S) from small ( F s ) via a dimensional transmutation, called "inverted hierarchy" mechanism. The first mechanism requires a flat direction which has a run-away behavior due to nonperturbative dynamics. If non-renormalizable terms lift the flat direction, the balance between non-perturbative and non-renormalizable terms in the potential produces a stable minimum, and one has a large (S) compared to the size of the potential, i.e. (Fs) = V 1/2 (see Fig. 3). Poppitz and Trivedi [8] presented a first example of this type of dynamics based on SU(13) x SU(ll). One of the problems in their model is that the scale (S /
291
turns out to be too large, and hence (Fs) must be also large to keep ( F s ) / ( S ) ~ 104 GeV. Then the gravitino mass m3/2 ,,~ ( F s ) / M p is large and we expect a large supergravity contribution to the scalar masses. The squark degeneracy, therefore, can be spoiled despite the fact that the model has GM contribution to the scalar masses; they are not the dominant contribution compared to the supergravity ones. The problem with the too-large supergravity contribution was resolved by employing a different model based on SU(7) x SU(6) gauge group which Nima Arkani-Hamed, John MarchRussell and myself proposed [9] (see also Erich Poppitz and Sandip Trivedi [10] in this proceedings). However, we realized (and also by Poppitz and Trivedi [11]) that there is a different serious problem with this model. There are several flat directions which are lifted only by a non-renormalizable operator and hence are light. Some of them are charged under the SSM gauge group. When they are stabilized by the nonrenormalizable operator and supersymmetry is broken with a large expectation value along a SSM gauge singlet direction among these flat directions, the SSM charged directions acquire supersymmetry breaking mass squared of order ( F s ) / ( S ) ,~ 104 GeV. Since this is a "large" supersymmetry breaking, such a large mass feeds into ordinary squark and slepton masses via twoloop renormalization group equations, and drive them negative. The question then is to find a model which does not have flat directions lifted only by non-renormalizable operators and charged under the SSM gauge group. I pursue the second possibility, "inverted hierarchy" [12] in this talk. This mechanism does not need an extra mass scale to be put in the model; both of the scales ( F s l and (S) are generated via dimensional transmutation. The idea is to have a classical flat direction S with an non-perturbatively induced linear term in the superpotential W = SA 2. It is then lifted only by the renormalization of the K~ihler potential: K = Z s S * S which modifies the potential V = A 4 / ( O 2 K / O S * O S ) . The one-loop gauge and Yukawa contributions to the Z s factor have opposite signs, and their balance can produce a min-
H. Murayama/NuclearPhysicsB (Proc. SuppL) 62A-C(1998) 289-298
292
case known to have the so-called quantum modified moduli space,
.4 1.2
.....~;!;iii!i!
pf(QiQj) = A2N+2. iii+ii +ii
r~
0.8
l
iii;Niiiiiiili +~;~+;~~ i~ i~ i~ ii
(1)
On the other hand, we can write a superpotential
Vl-loop
ii~ii!ii~i
0.6 ii!ii!ii!i
W = 1ASi~QiQJ
0.4
to couple Qi to SP(N) singlets Sij, whose supersymmetric extremum requires all meson operators to vanish: QiQj = 0 for any i,j. This is in a conflict with the quantum modified moduli space (1), and hence supersymmetry must be broken. Note that Sij are classically flat directions. The question then is where the vacuum is; is it at S = 0, S ~ A, or S >> A? The first two cases are genuinely strongly interacting and cannot be verified/excluded definitively. On the other hand, the last case S >> A is in a perturbative regime and can be checked whether it can be a consistent (local) minimum. Moreover this is the case of our interest to achieve the inverted hierarchy mechanism from the motivation discussed in the previous section. Let us concentrate on this case below; we will actually show that this cannot be a consistent minimum and S is driven back to the origin. The analysis is very simple. When S >> A, Qi acquire a large mass and one can integrate them out. Then the low-energy theory is a pure SP(N) Yang-MiUs theory which develops a gangino condensate, or a dynamical superpotential Weft = AeZ. S Note that the dynamical scale Ae# depends on S through the matching of the SP(N) gauge coupling at Q threshold, and hence can be rewritten in terms of the original SP(N) scale A
.......~,:,...........
0.2 0
i!iiiiii Iiiiii!ili:i !i;~ii i
~iiii!ii!ii~iiiiiiiL....,
10 o
...... ,
10 2
......., ......,... 10 e 101°
....... , ........ , ....... , ...... ~ ....... ,
10 4
10 6
S Figure 4. The scalar potential in models of inverted hierarchy. Both axes have arbitrary scales and are labeled only for the sake of emphasis that the horizontal scale is logarithmic.
imum of the potential. Since the Zs factor depends logarithmically on S, the minimum is naturally exponentially larger than the size of the potential, (S) >> (Fs) 1/2 as we would like (see Fig. 4). The situation is depicted in Fig. 4. To find a model of DSB with the inverted hierarchy, we need to study models which have a classical flat direction only lifted by the perturbative corrections to the K~.hler potential.
3. SP(N) M o d e l s Before explaining the model I propose, I review preliminaries of vector-like SP(N) DSB models by Izawa, Yanagida [13] and Intriligator, Thomas [14]. This model has the right feature for the inverted hierarchy mechanism: there is a classical flat direction which is lifted only by the renormalization of the K~hler potential. We will see, however, that the original version of this model cannot generate an exponentially large expectation value along the flat direction. We introduce quarks Qi in the fundamental representation (2N-dimensional) with N + 1 flavors (2N + 2 of them). There is hence a global SU(2N + 2) symmetry acting on Qi. This is the
(2)
as
Weft = (AS)A 2.
(3)
The KiLhler potential of the S field receives perturbative corrections from Q-loop at the one-loop level,
f d4OZs(~S)S*S.
(4I
Within the leading-log approximation, the wavefunction renormalization factor is [ SI 2
Zs ~,, 1 - 2N 1--~2 In M2
(5)
H. Murayama/NuclearPhysicsB (Proc. Suppl.) 62A-C(1998)289-298 1.4 1.2 1 ~'~ 0.8 . - ~ ,~.~ 0.6 ~
Wl -loop
I :~ i
1
0 . 4 ........ 0.2
~ii~i~
01 O0
i~ ++,,+,[
+ II+.,~
1 02
, t Itt+il[
I I l+q~l]
1 04
I l'q"~
i t ,,Sl~
10 6
I ll,u+~
+ ,++si~
t , ,,+,+d
.....
10 8 101°
S
Figure 5. Behavior of the potential in the vectorlike SP(N) models. The potential rises for large S and does not exhibit the inverted hierarchy mechanism.
where M is the ultraviolet cutoff. The effective potential long the S direction is given by
(
-llow
= \os*os]
~
IAh[2 x2 In IASI2/M 2 1 - 2N 1--g~
(6)
and is an increasing function of S, i.e. the field S is driven back to the origin (see Fig. 5). Therefore, this model does not allow a consistent potential minimum far away from the origin, as we wished.
293
hence they axe directly coupled to the dynamics of the supersymmetry breaking. We employ SP(4) gauge group with 5 flavors. As discussed earlier, we have Q+ with i = 1,2,-..,10, and the superpotential W = ½ASijQiQj. Into the global SU(10) symmetry, we embed SU(5)L x SU(5)n gauge group, and our SSM gauge groups axe further embedded into the diagonal subgroup of two SU(5)'s. (The SU(5)n group has only SU(3) x SU(2) x U(1) subgroup gauged. It may be that the SU(5)n "GUT" breaks down to SU(3) x SU(2) x U(1) at the GUT-scale, or the groups are unified in the sense of the string theory. We use SU(5)R language to keep the group theory as simple as possible.) Therefore, Qi and Sii decompose under the SP(4) x SU(5)L x SU(5)a gauge group as follows,
{ Sij--~
O
(8,
SU(5)L 5,
SU(5)n 1)
1, 5, 10,
5)
Sij
(8, (1, (1,
S'J
(1,
1,
10)
SP(4)
(7)
I)
Using this notation, the superpotential ASij QiQj also decomposes into three independent terms, 1
1
W = ~,QEQ + -~gSQQ + -~SQQ.
(8)
As discussed in the previous section, Sij = (E, S, S) are classical flat directions. Requiring vanishing D-terms under SU(5)L × SU(5)n, the classical flat direction of Sij is only in E direction:
4. T h e M o d e l Now let me present my model [15] (see also a similar model in [16]). The model is very similar to the SP(N) models discussed in the previous section but has a subgroup of the global SU(2N + 2) group gauged. This is the only essential difference, but it now allows a consistent minimum far away from the origin due to the inverted hierarchy mechanism. The hierarchy (S) >> (Fs) 1/2 is generated via a dimensional transmutation. The SSM gauge groups are embedded into the global SU(2N+2) symmetry and
1 =
1 1
(9) 1 1
Along this direction, SU(5)L x SU(5)n breaks down to their diagonal subgroup, which is "our" SU(5) gauge group. Note that the above particle content is completely chiral and none of the fields are allowed to have a mass term. There are no gauge singlet fields either. I find these features aesthetically appealing, and somewhat amusing
294
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
because the model is based on vector-like SP(N) models. Since we are interested in the inverted hierarchy (a> >> (Fa) 1/2, we can analyze the effective potential for a using the same technique as in the previous section, by first integrating out heavy Q, (~ and writing down the non-perturbative superpotential due to the SP(4) gaugino condensate. The only difference is in the K~ihler potential of a which now receives radiative corrections due to the SU(5)L x SU(5)R gauge interactions. We find - -, r^2~ ~ 'g2L~ Z~ = 1
+ •R °5
j) aj2 )d I) aJ 2 • In - - ~ - - 8 1 - - ~ 2 In - ~ - , ( 1 0 )
and the most important point is that the gauge contribution has the opposite sign from the Yukawa contribution. As a result, the potential can decrease for larger a when the gauge piece dominates over the Yukawa piece. Moreover, the gauge piece becomes smaller for larger a if the gauge group is asymptotically free while the Yukawa coupling can grow for larger a. Therefore, their relative magnitude can reverse at a critical value of a, beyond which the potential increases. Then the effective potential develops a consistent minimum. Since the minimum is determined by the balance between logarithmically running pieces, it is likely to be exponentially higher than the scale of supersymmetry breaking F~, hence the inverted hierarchy. Now let us briefly discuss the spectrum of this model. The SP(4) quarks (~ and Q become vector-like when the a breaks SU(5)L x SU(5)a to their diagonal subgroup, acquire a large mass A a / v ~ and decouple. Note that the a field has an F-component Fa = AA2 and hence Q, (~ are "messengers." It is interesting that the fields which have completely chiral quantum numbers can become vector-like messengers because of the dynamical breakdown of the gauge symmetry. The field a which acquires an expectation value is a part of the ~. multiplet, and other 24 components are all eaten by the broken SU(5) generators. The imaginary part of a is the Raxion in this model, and the real part is light: m~ ~ a/a ,,~ 100 GeV. Unfortunately the above particle content is not complete; to cancel the SU(5)~ and SU(5) 3
anomalies, we need more fields: SP(4)
¢4
(1, (1,
SU(5)L 5, 1,
SU(5)R
1) 5)
(i = 1,2).(11)
These fields acquire masses after the symmetry breaking through a non-renormalizable interaction W = ¢~4¢/M~. To keep these fields heavy enough ~> 100 GeV, we need the symmetry breaking scale to be high, a ~> 1014 GeV. Such a high scale may appear worrisome because the supergravity contribution to the scalar masses may become important as in the model of Ref. [8]. Fortunately, the GM contribution can be well dominant if a ~< 1016 GeV and hence there is a window for satisfying both requirements; this is a non-trivial success. Finally, S and S fields acquire masses of ,~ F~/a ,,, 10 TeV from the dynamical superpotential. If we want them to be heavier, one can introduce a non-renormalizable operator W = SE3S/M~. The detailed prediction on the gaugino and scalar mass spectrum is being worked out [17,18]. Let us summarize the important features of this model. • Welcome features: 1. No messenger sector. 2. No gauge singlets. 3. The particle content is completely chiral, and hence there is no need for forbidding mass terms by hand. 4. The inverted hierarchy mechanism works to make messenger fields heavy while keeping the squark, slepton masses at the weak scale. 5. The apparent unification of the gauge couplings is preserved. 6. All the scales are generated by the dimensional transmutation. 7. The dynamical breakdown of SU(5)L x SU(5)R --+ SU(5)v makes Q, ~) fields vector-like and they behave as messengers. • Undesirable features:
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
1. Anomaly canceling fields ¢i, ~i necessary, which requires a ~> 1014 GeV. But the supergravity contribution to scalar masses still under control for a ~< 1018 GeV. 2. We need to forbid an operator W = (detZ)/M 2 which could spoil the flatness of the a potential and hence the inverted hierarchy mechanism. 2
3. Re(a)
is analogous to the Polonyi-field in hidden sector models and is dangerous for cosmology. Fortunately, it has much stronger coupling than the Polonyi field (1/a vs. 1~MR), and hence is expected to decay before the nucleosynthesis [9]. Entropy production is still an issue (but see below).
4. The lightest neutralino, if abundant, decays into -y and gravitino, possibly after the nucleosynthesis and destroys light elements. This certainly has to be avoided. However, the entropy production from Re(a) decay dilutes it to the negligible level, as long as the lightest neutralino is not produced in its decay [20]. Affieck-Dine baryogenesis can be efficient enough to survive such an enormous entropy production [21]. Even though the last half of the above list describes certain undesirable features, they all appear surmountable, and I would say there are not too bad either, especially in comparison to the moduli-overclosure problem in low-energy GM models [21] or Polonyi problem in hidden sector models. An interesting aspect of this model is that one can let each of the SSM generations couple to either of the SU(5) factors; of course we would like to couple 1st and 2nd generations to the same SU(5) in order to generate degenerate scalar masses. ZOne way to justify it is to impose the anomalous U(1)R or its discrete subgroup, with charges Q(0), Q(0), S(2), :~(2), ~(2), ¢ i ( - 3 ) , ¢ i ( - 3 ) . This U(1)R symmetry is non-anomalous for the SP(4) group, but is likely to be anomalous for SU(5)L × SU(5)R, and Ira(a) is a potential candidate for the QCD axion with the decay constant in the interesting range [19].
295
5. Theoretical Subtlety I need to discuss briefly a theoretical subtlety concerning the discussion of the potential in the previous sections. I showed that the vector-like SP(N) models do not exhibit the inverted hierarchy mechanism in the absence of other gauge groups. However, the following argument seems to give the opposite conclusion. I will clarify what is incorrect in this argument to reassure the analysis in the previous sections. The (incorrect) argument is as follows. The a potential with the tree-level (canonical) K~ihler potential is just a constant,
(12)
Vtree = IAA212.
This potential can be "renormalization-group improved" by replacing the Yukawa coupling A by the running coupling constant A(a) [22], Virnproved = [A(a)A2[ 2 -
1 Z~ 1 IA°A212' Zs
(13)
where the coupling A0 is the bare coupling and the Z-factors are the coefficients in the K~iher potential of S and Q fields. Then the ZQ factor has a contribution from the SP(N) gauge interaction which h~s the opposite sign from the Yukawa contributions in Zs and ZQ. We would conclude that the inverted hierarchy mechanism occurs because of the competition between the gauge piece in ZQ and the Yulmwa piece in Zs and ZQ. To understand why the above argument is incorrect, we must go back to basics just as to resolve any other paradoxes. The Wilsonian Renormalization Group analysis goes as follows [23]. We start with a bare Lagrangian
+ f d48(QtoeVQo+StoSo)
(14)
where go2 (Ao) is the bare gauge (Yukawa) coupling at the ultraviolet cutoff M. As in any renormalization-group analysis, we begin the discussion with a dimensional analysis. The effective potential for the classical flat direction S is a function of these parameters,
V = V(S, g2,Ao;M)
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
296
:
e-4tV(etS, g2,Ao;etM)
(15)
where the last equality is the naive dimensional analysis which scales all dimensionful quantities by the same amount. The ultraviolet cutoff of the theory now is etM as a result. The next step is to "integrate out" modes between etM and M to bring the cutoff back to M. This procedure changes the bare coupling constants, and one finds
=
e-4tV'etSL ,go,'2 A'o;M).
(16)
Therefore, the S-dependence of the potential can be read off from the dependence on the running bare coupling constants. The important point in this analysis is the following. When we define the bare couplings g2 and Ao, we secretly assume that the bare fields are canonically renormalized. If they are not, one needs to include the normalization of the bare fields as another parameter of the Lagrangian. We stick to the usual convention not to include such an additional redundant parameter in the analysis. But this convention requires a rescaling of the S P ( N ) quarks Q after we "integrate out" modes between M and etM because the kinetic term receuves a radiative correction ZQ. And this rescaling actually contributes to the new bare coupling 1/g~2 because of an anomalous Jacobian. To be more explicit, the path integral on the field configuration space D ~ ~- T)QT)ST)V is divided into two momentum regions # < M and
M < # < etM,
the W W operator is exhausted at one-loop [23]: 1/g02 --~ 1/g~ + ~ t. When we rescale the matter fields to Q'o = ZQ/2Qo, S~ = Zls/2So, however, the path integral measure produces an anomalous Jacobian found by Konishi and Shizuya [24]. This fact is rather easy to understand. Since the matter fields are chiral superfields, one can generalize the rescaling factor to an arbitrary complex number. On the other hand, a phase rotation e i~ of a chiral superfield produces the ordinary chiral anomaly from the path integral measure: e x p ( - f d 2 6 ( N l / 4 r 2 ) i ~ W W + c.c.). Thanks to holomorphy, the phase dependence can be extended to an arbitrary complex number, and hence also to a real number (rescaling). After taking this Jacobian into account, the Lagrangian can be brought back to the conventional form
= f
J p
with new bare coupling constants: 8r 2 g~2
-
A'o =
e
(19)
Z~I(M, etM)Zsl/2(M, etM))~o.
(20)
A(etM, 1/g2o) - (etM)e s~2/g2b° 'must
be
t e --4t V(e t S,g ot2 ,~o; M):
- E (-~,,ko;M~
\ go
__g-~+b°t-NllnZQ(M'etM)'
The most important result is that the dynamical scale in the effective potential e - 4 t y (et S, g~ , A0; et M )
actually
/)~
8~r2
/
changed
(21) to
that
in
A'(M, 1/g~ 2) ----M e s~r2/g'°%° T)(I) e
= f #
L
X~,o
,,
,'
+ f d40(ZQ(M,e' M)Q?oeVQo+Z8(M,e' M)S?oSo)],
(17) and now the normalization of the K~ihler potential does not follow the convention, s Holomorphy requires that the change in coefficient of 3Here and below we suppress space-time integral and unimportant constants such as 1/4 in formulae to save space.
:
h(etM, 1/g2)ZQ Nf/b°.
(22)
Note that b0 = - 2 ( N + 1) and Nf = N + 1. The dynamical scale actually runs when the cutoff is changed! Surprising? Well, this is actually necessary to keep the consistency of all known exact results in supersymmetric gauge theories. Remember, for instance, the quantum modified moduli space Pf QiQj = A2N+2 in S P ( N ) theories. When one integrates out modes from the cutoff M down to M', we have another Lagrangian with canonically normalized
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
bare fields Q,i which satisfy Pf Q,iQ,j = A,2N+2. Since Q,i = ~t/2(~/r, .~Q ~.~ , M)Q i, the left-hand side of the constraint changes, so must the right-hand side. The required change is A' = Z~/2(M ', M)A which is nothing but the result obtained above. A similar analysis applies to the Affieck-DineSeiberg superpotential. For instance in SU(2) theory with one flavor, the superpotential is W = As/((~Q). Again with a different cutoff, the canonically normalized fields Q' and Q' appear in the superpotential W = A'5/(Q'Q'). For them to be identical, we need A' = .~Q ~1/l°71/l° .~) which is again consistent with the above general argument. Knowing the correct change in the bare parameters, now we can resolve the paradox. In vectorlike SP(N) models of DSB, the tree-level potential is Vtr~e =
IAA212.
(23)
The renormalization-group improvement now requires the replacement of both A and A by their running counterparts Vimproved
:
IA(a)A(a)212
=
-
A 1
I A I2
ZQ(a) A2 2
(24)
Then the potential does not depend on ZQ, and therefore not on the SP(N) gauge coupling. In this manner, we see that the analysis in the earlier section was indeed correct, and there is no inverted hierarchy mechanism in this model. On the other hand, the model I proposed has gauge interactions for the a field and Z~ alone has competing Yukawa and gauge pieces. 6. C o n c l u s i o n
I have presented the first phenomenologically viable model of dynamical supersymmetry breaking without a messenger sector. The model is simpler than the original models of gauge mediation, and I think it has some charm in it.
297
REFERENCES
1. E. Witten, Nucl. Phys. B188, 513 (1981). 2. K. Intriligator and N. Seiberg, Nucl. Phys. Proc. Suppl. 45BC, 1 (1996). 3. M. Dine and A. Nelson, Phys. Rev. D48, 1277 (1993). 4. M. Dine, A. Nelson, and Y. Shirman, Phys. Rev. D51, 1362 (1995); M. Dine, A. Nelson, Y. Nit, and Y. Shirman, Phys. Rev. D53, 2658 (1996). 5. M. Dine, W. Fischler, and M. Srednicki, Nucl. Phys. B189, 575 (1981); C. Nappi and B. Ovrut, Phys. Lett. B l 1 3 , 175 (1982); M. Dine and W. Fischler, Nucl. Phys. B204, 346 (1982); L. Alvarez-Gaumd, M. Claudson and M. Wise, Nucl. Phys. B207 96 (1982). 6. T. Hotta, K.-I. Izawa, and T. Yanagida, Report UT-752, hep-ph/9606203; N. Haba, N. Maru, T. Matsuoka, DPNU-96-63, hepph/9612468; L. Randall, MIT-CTP-2591, hep-ph/9612426; Y. Shadmi, Fermilab-Pub97/060-T, hep-ph/9703312; K.-I. Izawa, Y. Nomura, K. Tobe, and T. Yanagida, UT774, hep-ph/9705228. 7. A. Nelson, Talk given at 5th International Conference on Supersymmetries in Physics (SUSY 97), Philadelphia, PA, 27-31 May 1997. UW-PT-97-20, hep-ph/9707442 8. E. Poppitz and S. Trivedi, Fermilab Report EFI-96-35, hep-ph/9609529. 9. N. Arkani-Hamed, J. March-Russell, and H. Murayama, LBNL-39865, hep-ph/9701286. 10. E. Poppitz, Talk given at 5th International Conference on Supersymmetries in Physics (SUSY 97), Philadelphia, PA, 27-31 May 1997; E. Poppitz and S. Trivedi, hepph/9707439. 11. E. Poppitz and S.P. Trivedi, Phys. Lett. B401, 38 (1997). 12. E. Witten, Phys. Lett. 105B, 267 (1981). 13. K.-I. Izawa and T. Yanagida, hep-th/ 9602180, Prog. Theor. Phys. 95, 829 (1996). 14. K. Intriligator and S. Thomas, hep-th/ 9603158, Nuel. Phys. B473, 121 (1996). 15. H. Murayama, Phys. Rev. Lett. 79.18 (1997). 16. S. Dimopoulos, G. Dvali, R. tLuttazzi, and G.F. Giudice, CERN-TH-97-098, hep-
298
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ph/9705307. 17. G.F. Giudice, R. Rattazzi, CERN-TH-97-145, hep-ph/9706540. 18. C.D. Carone, L.J. Hall, T. Moroi, H. Murayama, and J. Terning, in preparation. 19. M. Kawasaki, T. Moroi, and T. Yanagida, Phys. gett. B383, 313 (1996). 20. T. Moroi, M. Yamaguchi, and T. Yanagida, Phys. Lett. B342, 105 (1995). 21. A. de GouvSa, T. Moroi, and H. Murayama, Phys. Rev. D56, 1281 (1997). 22. Y. Shirma~, Phys. Lett. B389 287 (1996). 23. N. Arkani-Hamed and H. Murayama, LBL40346, hep-th/9705189, to appear in Phys. Rev. D. 24. K. Konishi and K. Shizuya, Nuovo Gim. 90A, 111 (1985).
Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
Gauge Mediation Without A Messenger Sector Hitoshi Murayamaab* aDepartment of Physics, University of California Berkeley, CA 94720 bTheoretical Physics Group, Lawrence Berkeley National Laboratory University of California, Berkeley, CA 94720 I present a model of gauge mediation without a messenger sector. This is the first phenomenologically viable model of this type. The sector of dynamical supersymmetry breaking couples directly to the standard model gauge groups. The apparent gauge unification is preserved despite the direct coupling. The inverted hierarchy mechanism plays a crucial role in dynamics. The model is completely chiral, and hence there is no need to forbid mass terms by hand. Still vector-like messengers emerge as a consequence of dynamical breakdown of gauge symmetries.
1. I n t r o d u c t i o n
The dynamical supersymmetry breaking (DSB) [1] is an attractive idea to explain the hierarchy between the electroweak scale (or m z = 91.2 GeV) and the fundamental scale of nature M, possibly the GUT-scale (2 x 1016 GeV), the string scale (5 x 1017 GeV) or the reduced Planck scale (2 × 1018 GeV). The electroweak scale is protected by the supersymmetry up to all orders in perturbation theory but is generated by nonperturbative effects through dimensional transmutation, m z ", M e x p ( 8 7 r 2 / g 2 ( M ) b o ) << M with an asymptotically free gauge theory bo < 0. However, the implementation of this idea has been impeded by the technical difficulties of model building and the lack of understanding in non-perturbative dynamics of supersymmetric gauge theories. Since the latter problem has been largely resolved by recent works of Seiberg and his collaborators (for a review, see e.g., [2]), it is a pressing question whether one can build realistic models. *This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC0376SF00098 and in part by the National Science Foundation under grant PHY-90-21139, and also by Alfred P. Sloan Foundation. 0920-5632/98/$19.00 1998 Elsevier Science B.V. PII S0920-5632(97)00670-1
It was the pioneering works by Dine and Nelson [3] (later together with Shirman and Nir [4]) which demonstrated that one can implement the DSB in a phenomenologically viable manner, by means of gauge mediation (GM) [5]. The models presented in these works had two major important features: the hierarchy explained by the DSB and an automatic squark degeneracy thanks to the GM. Unfortunately the structure of the models was quite complicated. There are three sectors in the models which are "insulated" from each other: the DSB sector, the "messenger sector," and the supersymmetric standard model (SSM); see Fig. 1. The need for the DSB and SSM sectors is obvious; what bothered me is the apparent lack of motivation for the messenger sector. There are a couple of awkward features in the messenger sector: (1) It has to be put in by hand for no other reason. (2) It has to be "insulated" from the other two sectors in the superpotential, with only interactions being the messenger U(1) to the DSB sector and the SSM gauge interactions. (3) The "messenger" fields are vector-like under the SSM gauge interactions while their mass terms have to be forbidden. (4) Fields which are completely gauge singlets play major roles in the GM. The claim in this talk is that one can eliminate the messenger sector entirely (see [6] for other attempts to simplify the GM models). The DSB
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
290
IO,O00TeV
DSB
~
lOOTeV
1 TeV
100 GeV
DSB
Sector
? TeV
Sector
mes$
IMessenger Sector
1 TeV 100 GeV
SSM Sector
Figure 1. Structure of the original models of gauge mediation. The label GSM stands for the standard model gauge groups, and U(1)mess for the messenger U(1) gauge group. The energy scales are only approximate.
sector and SSM sector are directly coupled to each other (see Fig. 2) and the model I present is the first phenomenologically viable model of this type. It moreover has several nice features: (1) The model is completely chiral and there is no need to drop mass terms by hand. (2) There are no gange-singlet fields. (3) The apparent perturbative unification of gauge coupling constants is preserved. 2. P r o b l e m s w i t h D i r e c t G a u g e M e d i a t i o n
In the past year, there has been an intense activity to build models of GM in which the DSB sector and SSM sector are directly coupled via SSM gauge interactions and hence there is no need for a separate messenger sector (see Ann Nelson in this proceedings [7]). Of course, this is probably the most natural way to build the GM models and was indeed discussed already a decade ago. However, there used to be a major difficulty with this idea. If two sectors are directly coupled through the SSM gauge interactions, the DSB sector must have a large (un: broken) global symmetry which can accommo-
SSM Sector
Figure 2. Structure of the gauge mediation model without a messenger sector. The energy scale of DSB is model-dependent.
date the full SSM gauge group (hopefully even the GUT-gauge group such as SU(5) or SO(10)). The smallest DSB model which was known to accommodate SU(5) global symmetry back then has the dynamical gauge group SU(15), which results in 15 additional fields in 5 representation of SU(5) and lets all SSM gauge couplings to reach their Landau poles below 107 GeV. Even though there is nothing a priori wrong with such a behavior of gauge coupling constants, especially in view of the duality among the supersymmetry gauge theories, it is disheartening to throw away the apparent gauge unification which hints towards supersymmetric GUT or string unification. This problem can be solved if the DSB sector has several mass scales. The essential ingredient in this approach is that there is a field S which generates masses for the (effective) messengers Q, Q through the superpotential coupling W = XSQQ with very different sizes for the expectation values for A- and F-components, (S) 2 >> (Fs). The size of supersymmetry breaking effects in the SSM is characterized by the ratio ( F s ) / ( S ) and it must be kept around 104 GeV. In the original attempt of the direct GM, they are comparable in magnitudes: (S) ,~ (Fs) 1/2 ,,~ 104 GeV, and one finds that there are many messengers (15 of them) at 104 GeV which make all the SSM gauge coupling constants blow up quickly. On the other hand, if (S) >> (Fs) 1/2, the messenger fields can be heavy enough not to
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
0.8 gnon_pert
r.~
i
0.6
~'~ 0.4
'\
0
i
20
40
S
60
80
100
Figure 3. The scalar potential in models with a flat direction lifted only by a non-renormalizable operator. Both axes have arbitrary scales and are labeled only for the sake of emphasis that the horizontal scale is linear.
make the gauge coupling constants blow up while keeping the size of scalar, gaugino masses in the SSM sector at the electroweak scale. For instance, if (Fs) ~ (10 9 GeV) 2 and (S) ~ 1014 GeV, there is basically no concern about the Landau poles while the size of the superparticle masses comes out correctly. The very question then is how to generate the hierarchy (S) >> (Fs) 1/2. There are at least two logical possibilities. One is to employ a model which already has a mass scale built in, such as non-renormalizable models with Planck-scale suppressed operators. The other is to generate a large (S) from small ( F s ) via a dimensional transmutation, called "inverted hierarchy" mechanism. The first mechanism requires a flat direction which has a run-away behavior due to nonperturbative dynamics. If non-renormalizable terms lift the flat direction, the balance between non-perturbative and non-renormalizable terms in the potential produces a stable minimum, and one has a large (S) compared to the size of the potential, i.e. (Fs) = V 1/2 (see Fig. 3). Poppitz and Trivedi [8] presented a first example of this type of dynamics based on SU(13) x SU(ll). One of the problems in their model is that the scale (S /
291
turns out to be too large, and hence (Fs) must be also large to keep ( F s ) / ( S ) ~ 104 GeV. Then the gravitino mass m3/2 ,,~ ( F s ) / M p is large and we expect a large supergravity contribution to the scalar masses. The squark degeneracy, therefore, can be spoiled despite the fact that the model has GM contribution to the scalar masses; they are not the dominant contribution compared to the supergravity ones. The problem with the too-large supergravity contribution was resolved by employing a different model based on SU(7) x SU(6) gauge group which Nima Arkani-Hamed, John MarchRussell and myself proposed [9] (see also Erich Poppitz and Sandip Trivedi [10] in this proceedings). However, we realized (and also by Poppitz and Trivedi [11]) that there is a different serious problem with this model. There are several flat directions which are lifted only by a non-renormalizable operator and hence are light. Some of them are charged under the SSM gauge group. When they are stabilized by the nonrenormalizable operator and supersymmetry is broken with a large expectation value along a SSM gauge singlet direction among these flat directions, the SSM charged directions acquire supersymmetry breaking mass squared of order ( F s ) / ( S ) ,~ 104 GeV. Since this is a "large" supersymmetry breaking, such a large mass feeds into ordinary squark and slepton masses via twoloop renormalization group equations, and drive them negative. The question then is to find a model which does not have flat directions lifted only by non-renormalizable operators and charged under the SSM gauge group. I pursue the second possibility, "inverted hierarchy" [12] in this talk. This mechanism does not need an extra mass scale to be put in the model; both of the scales ( F s l and (S) are generated via dimensional transmutation. The idea is to have a classical flat direction S with an non-perturbatively induced linear term in the superpotential W = SA 2. It is then lifted only by the renormalization of the K~ihler potential: K = Z s S * S which modifies the potential V = A 4 / ( O 2 K / O S * O S ) . The one-loop gauge and Yukawa contributions to the Z s factor have opposite signs, and their balance can produce a min-
H. Murayama/NuclearPhysicsB (Proc. SuppL) 62A-C(1998) 289-298
292
case known to have the so-called quantum modified moduli space,
.4 1.2
.....~;!;iii!i!
pf(QiQj) = A2N+2. iii+ii +ii
r~
0.8
l
iii;Niiiiiiili +~;~+;~~ i~ i~ i~ ii
(1)
On the other hand, we can write a superpotential
Vl-loop
ii~ii!ii~i
0.6 ii!ii!ii!i
W = 1ASi~QiQJ
0.4
to couple Qi to SP(N) singlets Sij, whose supersymmetric extremum requires all meson operators to vanish: QiQj = 0 for any i,j. This is in a conflict with the quantum modified moduli space (1), and hence supersymmetry must be broken. Note that Sij are classically flat directions. The question then is where the vacuum is; is it at S = 0, S ~ A, or S >> A? The first two cases are genuinely strongly interacting and cannot be verified/excluded definitively. On the other hand, the last case S >> A is in a perturbative regime and can be checked whether it can be a consistent (local) minimum. Moreover this is the case of our interest to achieve the inverted hierarchy mechanism from the motivation discussed in the previous section. Let us concentrate on this case below; we will actually show that this cannot be a consistent minimum and S is driven back to the origin. The analysis is very simple. When S >> A, Qi acquire a large mass and one can integrate them out. Then the low-energy theory is a pure SP(N) Yang-MiUs theory which develops a gangino condensate, or a dynamical superpotential Weft = AeZ. S Note that the dynamical scale Ae# depends on S through the matching of the SP(N) gauge coupling at Q threshold, and hence can be rewritten in terms of the original SP(N) scale A
.......~,:,...........
0.2 0
i!iiiiii Iiiiii!ili:i !i;~ii i
~iiii!ii!ii~iiiiiiiL....,
10 o
...... ,
10 2
......., ......,... 10 e 101°
....... , ........ , ....... , ...... ~ ....... ,
10 4
10 6
S Figure 4. The scalar potential in models of inverted hierarchy. Both axes have arbitrary scales and are labeled only for the sake of emphasis that the horizontal scale is logarithmic.
imum of the potential. Since the Zs factor depends logarithmically on S, the minimum is naturally exponentially larger than the size of the potential, (S) >> (Fs) 1/2 as we would like (see Fig. 4). The situation is depicted in Fig. 4. To find a model of DSB with the inverted hierarchy, we need to study models which have a classical flat direction only lifted by the perturbative corrections to the K~.hler potential.
3. SP(N) M o d e l s Before explaining the model I propose, I review preliminaries of vector-like SP(N) DSB models by Izawa, Yanagida [13] and Intriligator, Thomas [14]. This model has the right feature for the inverted hierarchy mechanism: there is a classical flat direction which is lifted only by the renormalization of the K~hler potential. We will see, however, that the original version of this model cannot generate an exponentially large expectation value along the flat direction. We introduce quarks Qi in the fundamental representation (2N-dimensional) with N + 1 flavors (2N + 2 of them). There is hence a global SU(2N + 2) symmetry acting on Qi. This is the
(2)
as
Weft = (AS)A 2.
(3)
The KiLhler potential of the S field receives perturbative corrections from Q-loop at the one-loop level,
f d4OZs(~S)S*S.
(4I
Within the leading-log approximation, the wavefunction renormalization factor is [ SI 2
Zs ~,, 1 - 2N 1--~2 In M2
(5)
H. Murayama/NuclearPhysicsB (Proc. Suppl.) 62A-C(1998)289-298 1.4 1.2 1 ~'~ 0.8 . - ~ ,~.~ 0.6 ~
Wl -loop
I :~ i
1
0 . 4 ........ 0.2
~ii~i~
01 O0
i~ ++,,+,[
+ II+.,~
1 02
, t Itt+il[
I I l+q~l]
1 04
I l'q"~
i t ,,Sl~
10 6
I ll,u+~
+ ,++si~
t , ,,+,+d
.....
10 8 101°
S
Figure 5. Behavior of the potential in the vectorlike SP(N) models. The potential rises for large S and does not exhibit the inverted hierarchy mechanism.
where M is the ultraviolet cutoff. The effective potential long the S direction is given by
(
-llow
= \os*os]
~
IAh[2 x2 In IASI2/M 2 1 - 2N 1--g~
(6)
and is an increasing function of S, i.e. the field S is driven back to the origin (see Fig. 5). Therefore, this model does not allow a consistent potential minimum far away from the origin, as we wished.
293
hence they axe directly coupled to the dynamics of the supersymmetry breaking. We employ SP(4) gauge group with 5 flavors. As discussed earlier, we have Q+ with i = 1,2,-..,10, and the superpotential W = ½ASijQiQj. Into the global SU(10) symmetry, we embed SU(5)L x SU(5)n gauge group, and our SSM gauge groups axe further embedded into the diagonal subgroup of two SU(5)'s. (The SU(5)n group has only SU(3) x SU(2) x U(1) subgroup gauged. It may be that the SU(5)n "GUT" breaks down to SU(3) x SU(2) x U(1) at the GUT-scale, or the groups are unified in the sense of the string theory. We use SU(5)R language to keep the group theory as simple as possible.) Therefore, Qi and Sii decompose under the SP(4) x SU(5)L x SU(5)a gauge group as follows,
{ Sij--~
O
(8,
SU(5)L 5,
SU(5)n 1)
1, 5, 10,
5)
Sij
(8, (1, (1,
S'J
(1,
1,
10)
SP(4)
(7)
I)
Using this notation, the superpotential ASij QiQj also decomposes into three independent terms, 1
1
W = ~,QEQ + -~gSQQ + -~SQQ.
(8)
As discussed in the previous section, Sij = (E, S, S) are classical flat directions. Requiring vanishing D-terms under SU(5)L × SU(5)n, the classical flat direction of Sij is only in E direction:
4. T h e M o d e l Now let me present my model [15] (see also a similar model in [16]). The model is very similar to the SP(N) models discussed in the previous section but has a subgroup of the global SU(2N + 2) group gauged. This is the only essential difference, but it now allows a consistent minimum far away from the origin due to the inverted hierarchy mechanism. The hierarchy (S) >> (Fs) 1/2 is generated via a dimensional transmutation. The SSM gauge groups are embedded into the global SU(2N+2) symmetry and
1 =
1 1
(9) 1 1
Along this direction, SU(5)L x SU(5)n breaks down to their diagonal subgroup, which is "our" SU(5) gauge group. Note that the above particle content is completely chiral and none of the fields are allowed to have a mass term. There are no gauge singlet fields either. I find these features aesthetically appealing, and somewhat amusing
294
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
because the model is based on vector-like SP(N) models. Since we are interested in the inverted hierarchy (a> >> (Fa) 1/2, we can analyze the effective potential for a using the same technique as in the previous section, by first integrating out heavy Q, (~ and writing down the non-perturbative superpotential due to the SP(4) gaugino condensate. The only difference is in the K~ihler potential of a which now receives radiative corrections due to the SU(5)L x SU(5)R gauge interactions. We find - -, r^2~ ~ 'g2L~ Z~ = 1
+ •R °5
j) aj2 )d I) aJ 2 • In - - ~ - - 8 1 - - ~ 2 In - ~ - , ( 1 0 )
and the most important point is that the gauge contribution has the opposite sign from the Yukawa contribution. As a result, the potential can decrease for larger a when the gauge piece dominates over the Yukawa piece. Moreover, the gauge piece becomes smaller for larger a if the gauge group is asymptotically free while the Yukawa coupling can grow for larger a. Therefore, their relative magnitude can reverse at a critical value of a, beyond which the potential increases. Then the effective potential develops a consistent minimum. Since the minimum is determined by the balance between logarithmically running pieces, it is likely to be exponentially higher than the scale of supersymmetry breaking F~, hence the inverted hierarchy. Now let us briefly discuss the spectrum of this model. The SP(4) quarks (~ and Q become vector-like when the a breaks SU(5)L x SU(5)a to their diagonal subgroup, acquire a large mass A a / v ~ and decouple. Note that the a field has an F-component Fa = AA2 and hence Q, (~ are "messengers." It is interesting that the fields which have completely chiral quantum numbers can become vector-like messengers because of the dynamical breakdown of the gauge symmetry. The field a which acquires an expectation value is a part of the ~. multiplet, and other 24 components are all eaten by the broken SU(5) generators. The imaginary part of a is the Raxion in this model, and the real part is light: m~ ~ a/a ,,~ 100 GeV. Unfortunately the above particle content is not complete; to cancel the SU(5)~ and SU(5) 3
anomalies, we need more fields: SP(4)
¢4
(1, (1,
SU(5)L 5, 1,
SU(5)R
1) 5)
(i = 1,2).(11)
These fields acquire masses after the symmetry breaking through a non-renormalizable interaction W = ¢~4¢/M~. To keep these fields heavy enough ~> 100 GeV, we need the symmetry breaking scale to be high, a ~> 1014 GeV. Such a high scale may appear worrisome because the supergravity contribution to the scalar masses may become important as in the model of Ref. [8]. Fortunately, the GM contribution can be well dominant if a ~< 1016 GeV and hence there is a window for satisfying both requirements; this is a non-trivial success. Finally, S and S fields acquire masses of ,~ F~/a ,,, 10 TeV from the dynamical superpotential. If we want them to be heavier, one can introduce a non-renormalizable operator W = SE3S/M~. The detailed prediction on the gaugino and scalar mass spectrum is being worked out [17,18]. Let us summarize the important features of this model. • Welcome features: 1. No messenger sector. 2. No gauge singlets. 3. The particle content is completely chiral, and hence there is no need for forbidding mass terms by hand. 4. The inverted hierarchy mechanism works to make messenger fields heavy while keeping the squark, slepton masses at the weak scale. 5. The apparent unification of the gauge couplings is preserved. 6. All the scales are generated by the dimensional transmutation. 7. The dynamical breakdown of SU(5)L x SU(5)R --+ SU(5)v makes Q, ~) fields vector-like and they behave as messengers. • Undesirable features:
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
1. Anomaly canceling fields ¢i, ~i necessary, which requires a ~> 1014 GeV. But the supergravity contribution to scalar masses still under control for a ~< 1018 GeV. 2. We need to forbid an operator W = (detZ)/M 2 which could spoil the flatness of the a potential and hence the inverted hierarchy mechanism. 2
3. Re(a)
is analogous to the Polonyi-field in hidden sector models and is dangerous for cosmology. Fortunately, it has much stronger coupling than the Polonyi field (1/a vs. 1~MR), and hence is expected to decay before the nucleosynthesis [9]. Entropy production is still an issue (but see below).
4. The lightest neutralino, if abundant, decays into -y and gravitino, possibly after the nucleosynthesis and destroys light elements. This certainly has to be avoided. However, the entropy production from Re(a) decay dilutes it to the negligible level, as long as the lightest neutralino is not produced in its decay [20]. Affieck-Dine baryogenesis can be efficient enough to survive such an enormous entropy production [21]. Even though the last half of the above list describes certain undesirable features, they all appear surmountable, and I would say there are not too bad either, especially in comparison to the moduli-overclosure problem in low-energy GM models [21] or Polonyi problem in hidden sector models. An interesting aspect of this model is that one can let each of the SSM generations couple to either of the SU(5) factors; of course we would like to couple 1st and 2nd generations to the same SU(5) in order to generate degenerate scalar masses. ZOne way to justify it is to impose the anomalous U(1)R or its discrete subgroup, with charges Q(0), Q(0), S(2), :~(2), ~(2), ¢ i ( - 3 ) , ¢ i ( - 3 ) . This U(1)R symmetry is non-anomalous for the SP(4) group, but is likely to be anomalous for SU(5)L × SU(5)R, and Ira(a) is a potential candidate for the QCD axion with the decay constant in the interesting range [19].
295
5. Theoretical Subtlety I need to discuss briefly a theoretical subtlety concerning the discussion of the potential in the previous sections. I showed that the vector-like SP(N) models do not exhibit the inverted hierarchy mechanism in the absence of other gauge groups. However, the following argument seems to give the opposite conclusion. I will clarify what is incorrect in this argument to reassure the analysis in the previous sections. The (incorrect) argument is as follows. The a potential with the tree-level (canonical) K~ihler potential is just a constant,
(12)
Vtree = IAA212.
This potential can be "renormalization-group improved" by replacing the Yukawa coupling A by the running coupling constant A(a) [22], Virnproved = [A(a)A2[ 2 -
1 Z~ 1 IA°A212' Zs
(13)
where the coupling A0 is the bare coupling and the Z-factors are the coefficients in the K~iher potential of S and Q fields. Then the ZQ factor has a contribution from the SP(N) gauge interaction which h~s the opposite sign from the Yukawa contributions in Zs and ZQ. We would conclude that the inverted hierarchy mechanism occurs because of the competition between the gauge piece in ZQ and the Yulmwa piece in Zs and ZQ. To understand why the above argument is incorrect, we must go back to basics just as to resolve any other paradoxes. The Wilsonian Renormalization Group analysis goes as follows [23]. We start with a bare Lagrangian
+ f d48(QtoeVQo+StoSo)
(14)
where go2 (Ao) is the bare gauge (Yukawa) coupling at the ultraviolet cutoff M. As in any renormalization-group analysis, we begin the discussion with a dimensional analysis. The effective potential for the classical flat direction S is a function of these parameters,
V = V(S, g2,Ao;M)
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
296
:
e-4tV(etS, g2,Ao;etM)
(15)
where the last equality is the naive dimensional analysis which scales all dimensionful quantities by the same amount. The ultraviolet cutoff of the theory now is etM as a result. The next step is to "integrate out" modes between etM and M to bring the cutoff back to M. This procedure changes the bare coupling constants, and one finds
=
e-4tV'etSL ,go,'2 A'o;M).
(16)
Therefore, the S-dependence of the potential can be read off from the dependence on the running bare coupling constants. The important point in this analysis is the following. When we define the bare couplings g2 and Ao, we secretly assume that the bare fields are canonically renormalized. If they are not, one needs to include the normalization of the bare fields as another parameter of the Lagrangian. We stick to the usual convention not to include such an additional redundant parameter in the analysis. But this convention requires a rescaling of the S P ( N ) quarks Q after we "integrate out" modes between M and etM because the kinetic term receuves a radiative correction ZQ. And this rescaling actually contributes to the new bare coupling 1/g~2 because of an anomalous Jacobian. To be more explicit, the path integral on the field configuration space D ~ ~- T)QT)ST)V is divided into two momentum regions # < M and
M < # < etM,
the W W operator is exhausted at one-loop [23]: 1/g02 --~ 1/g~ + ~ t. When we rescale the matter fields to Q'o = ZQ/2Qo, S~ = Zls/2So, however, the path integral measure produces an anomalous Jacobian found by Konishi and Shizuya [24]. This fact is rather easy to understand. Since the matter fields are chiral superfields, one can generalize the rescaling factor to an arbitrary complex number. On the other hand, a phase rotation e i~ of a chiral superfield produces the ordinary chiral anomaly from the path integral measure: e x p ( - f d 2 6 ( N l / 4 r 2 ) i ~ W W + c.c.). Thanks to holomorphy, the phase dependence can be extended to an arbitrary complex number, and hence also to a real number (rescaling). After taking this Jacobian into account, the Lagrangian can be brought back to the conventional form
= f
J p
with new bare coupling constants: 8r 2 g~2
-
A'o =
e
(19)
Z~I(M, etM)Zsl/2(M, etM))~o.
(20)
A(etM, 1/g2o) - (etM)e s~2/g2b° 'must
be
t e --4t V(e t S,g ot2 ,~o; M):
- E (-~,,ko;M~
\ go
__g-~+b°t-NllnZQ(M'etM)'
The most important result is that the dynamical scale in the effective potential e - 4 t y (et S, g~ , A0; et M )
actually
/)~
8~r2
/
changed
(21) to
that
in
A'(M, 1/g~ 2) ----M e s~r2/g'°%° T)(I) e
= f #
L
X~,o
,,
,'
+ f d40(ZQ(M,e' M)Q?oeVQo+Z8(M,e' M)S?oSo)],
(17) and now the normalization of the K~ihler potential does not follow the convention, s Holomorphy requires that the change in coefficient of 3Here and below we suppress space-time integral and unimportant constants such as 1/4 in formulae to save space.
:
h(etM, 1/g2)ZQ Nf/b°.
(22)
Note that b0 = - 2 ( N + 1) and Nf = N + 1. The dynamical scale actually runs when the cutoff is changed! Surprising? Well, this is actually necessary to keep the consistency of all known exact results in supersymmetric gauge theories. Remember, for instance, the quantum modified moduli space Pf QiQj = A2N+2 in S P ( N ) theories. When one integrates out modes from the cutoff M down to M', we have another Lagrangian with canonically normalized
H. Murayama/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 289-298
bare fields Q,i which satisfy Pf Q,iQ,j = A,2N+2. Since Q,i = ~t/2(~/r, .~Q ~.~ , M)Q i, the left-hand side of the constraint changes, so must the right-hand side. The required change is A' = Z~/2(M ', M)A which is nothing but the result obtained above. A similar analysis applies to the Affieck-DineSeiberg superpotential. For instance in SU(2) theory with one flavor, the superpotential is W = As/((~Q). Again with a different cutoff, the canonically normalized fields Q' and Q' appear in the superpotential W = A'5/(Q'Q'). For them to be identical, we need A' = .~Q ~1/l°71/l° .~) which is again consistent with the above general argument. Knowing the correct change in the bare parameters, now we can resolve the paradox. In vectorlike SP(N) models of DSB, the tree-level potential is Vtr~e =
IAA212.
(23)
The renormalization-group improvement now requires the replacement of both A and A by their running counterparts Vimproved
:
IA(a)A(a)212
=
-
A 1
I A I2
ZQ(a) A2 2
(24)
Then the potential does not depend on ZQ, and therefore not on the SP(N) gauge coupling. In this manner, we see that the analysis in the earlier section was indeed correct, and there is no inverted hierarchy mechanism in this model. On the other hand, the model I proposed has gauge interactions for the a field and Z~ alone has competing Yukawa and gauge pieces. 6. C o n c l u s i o n
I have presented the first phenomenologically viable model of dynamical supersymmetry breaking without a messenger sector. The model is simpler than the original models of gauge mediation, and I think it has some charm in it.
297
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