MCTP-03-56

Gravity-assisted exact unification in minimal

supersymmetric and its gaugino mass spectrum

Kazuhiro Tobe, James D. Wells

Michigan Center for Theoretical Physics (MCTP)

University of Michigan, Ann Arbor, MI 48109-1120, USA

Department of Physics, University of California, Davis 95616, USA

Minimal supersymmetric with exact unification is naively inconsistent with proton decay constraints. However, it can be made viable by a gravity-induced non-renormalizable operator connecting the adjoint Higgs boson and adjoint vector boson representations. We compute the allowed coupling space for this theory and find natural compatibility with proton decay constraints even for relatively light superpartner masses. The modifications away from the naive theory have an impact on the gaugino mass spectrum, which we calculate. A combination of precision Linear Collider and Large Hadron Collider measurements of superpartner masses would enable interesting tests of the high-scale form of minimal supersymmetric .

hep-ph/0312159

December 2003

The three gauge couplings of the minimal supersymmetric standard model (MSSM) unify to within 1% of each other at a high scale . Simple Grand Unified Theories (GUTs) predict such an outcome, where the low-scale gauge couplings must flow to within a small neighborhood of each other (less than few percent mismatch) at the high scale. Exact unification occurs only when all threshold corrections at the high scale are properly taken into account.

The simplest supersymmetric GUT model is minimal , with matter representations , the gauge boson representation , and Higgs representations . Precise gauge coupling unification at the high-scale must take into account threshold corrections from heavy GUT remnants of the , , and representations. The colored Higgsino triplets from the representations also contribute to dangerous dimension five operators mediating proton decay [1]. A careful analysis of both gauge coupling unification and proton decay in minimal supersymmetric concludes

(1) | |||||

(2) |

if superpartner masses are in the TeV region. This has led to the
conclusion that minimal is
dead [2, 3, 4]
or perhaps at least highly constrained with superpartner masses in the
range [5] which strains its ability
to naturally explain the electroweak symmetry breaking scale.
^{1}^{1}1There exist other models that are consistent
with both gauge coupling unification and proton decay. For example,
see Ref. [6].

In this letter we wish to point out two conclusions we have come to recently, apropos to the discussion above. First, similar to the effects found in Refs. [7]-[13] we have found that expected non-renormalizable gauge-kinetic operators in the GUT theory can redeem minimal without requiring unnaturally large coefficients. Second, we have computed the imprint of this effect on the gaugino masses, and found the resulting magnitudes of their relative shifts at the GUT scale to be within the sensitivities of future and planned colliders.

Minimal as a purely renormalizable supersymmetric theory was never viable because unification of down-quark Yukawa couplings with lepton Yukawa couplings does not work for the first two generations. It has been understood for a very long time now that non-trivial non-renormalizable operators (NROs) are needed. This is no extraordinary burden on the theory, however, as the Planck scale is not far from the GUT scale and NROs induced by supergravity are expected and of sufficient size to implement the flavor gymnastics required to reproduce the masses and mixings of the quarks and leptons.

It has also been known for some time that NROs can dramatically affect gauge coupling unification and gaugino masses [7]-[13]. These operators should not necessarily be viewed as sources of GUT-scale obfuscation, but rather as potential saviors for a theory that struggles to survive without them. Minimal is one such theory.

We can write the gauge-kinetic function of minimal as

(3) |

where and contains the effective singlet supersymmetry breaking. The gauge coupling is and the universal contribution to the masses of all gauginos is .

This second term of Eq. (3) is the focus of our
analysis^{2}^{2}2See Refs. [5, 14]
for discussion of other types of NROs useful to cure the minimal
problem. as it connects the adjoint Higgs representation
to the gauge fields via a NRO. Not only is the operator expected,
but it is guaranteed to contribute to the gauge coupling corrections
because the adjoint Higgs must get a vacuum expectation value (vev)
of the form

(4) |

to break to at the GUT scale. The numerical value of depends on details of the couplings but should be around the GUT scale of .

The relationships between the GUT scale gauge coupling and the low-scale gauge couplings of the MSSM effective theory are

(5) |

where and for the gauge groups respectively. Here we adopt the GUT normalized gauge coupling . The functions are the threshold corrections due to heavy GUT states; where and are function coefficient of a heavy particle and its mass, respectively. They are explicitly written by

(6) | |||||

(7) | |||||

(8) |

We will be working with this equation near the GUT scale, , and so the couplings are assumed to be those that have been measured at the weak scale, renormalized by weak-scale supersymmetric particle threshold corrections and run up to the high scale using two-loop renormalization group evolution [15]. Our equations are always in the scheme.

It is easy to see how the triplet Higgsino mass is severely constrained by unification requirements. Let’s consider the case for a moment. There exists a linear combination of that depends only on and not on the other unknown GUT scale states [16]:

(9) |

This equation is true for any scale at the one-loop level, but it is most instructive to evaluate it at the unification scale , which we define to be the place where ,

(10) |

depends mildly on the low-scale superpartner masses, but it is always within the range

(11) |

for superpartner masses at the TeV scale and below.

It is well known [15] that , albeit by less than 1%. Nevertheless, this implies that the LHS of Eq. (10) is necessarily negative. We see that is required for the RHS to be negative and successful gauge coupling unification to occur. But this is in conflict with the proton decay requirement that .

However, non-zero can easily and naturally
enable a large .
Because of an interesting and non-trivial relation between
and a -function coefficients of
(), an inclusion of non-zero
only affects the unified gauge coupling and color-triplet
Higgsino mass as can be seen from Eq. (5). In other
words, if we define
the effective colored Higgsino mass to be
, the above
constraints discussion
in the case with applies to .
Therefore, even though the effective
colored Higgsino mass is severely constrained by
gauge coupling unification (Eq. (2)), the real
colored Higgsino mass can be large enough to satisfy the proton decay limit
Eq. (1) if is positive and
of order a few percent.^{3}^{3}3
Other heavy particle masses are constrained by the gauge coupling
unification as GeV GeV. However, this constraint does not change
even if non-zero is taken into account because of
the relation .
We remark also that the unified coupling governing dimension six
proton decay operators is reduced by , thus increasing the proton lifetime.

We have done the precise numerical work to test this supposition and the results are presented in Fig. 1, where the relationship between and for exact unification is established. Each band is for a given assumed superpartner spectrum, and the width of the band is primarily due to the current uncertainty in which we take to be .

We see from the numerical results (Fig. 1) that if we ignore the adjoint-Higgs NRO correction () the triplet Higgsino mass needed for unification is less than about , even for all superpartner masses up to . However, if we find that can be comfortably greater than , thus enabling precision gauge coupling unification and a sufficiently long-lived proton. This successful region of parameter space requires , which is consistent with the expectation . It should be stressed that this is a built-in mechanism to increase naturally in minimal model, and more generally, in models in which the breaks .

At present there is no known way to experimentally verify minimal , or any other GUT for that matter. However, it is possible to test the theory nontrivially. On the surface it may appear unlikely that any shifting around of and at the high-scale to obtain compatibility with low-scale gauge coupling measurements would have any discernible experimental implications. However, precision gaugino mass measurements do provide a interesting probe of the framework.

One crucial realization is that the representation vev is not just a scalar vev, but a vev in superspace when we take into account the entire chiral superfield,

(12) |

Just as the and Higgs superfields of the MSSM pick up auxiliary field vevs when their scalar components condense, the superfield obtains a superspace vev.

The superpotential and soft lagrangian terms we assume are

(13) |

(14) |

where upon minimizing the full potential we find

(15) |

which generates a correction to gaugino masses via the NRO in Eq. (3).

This non-zero -term vev contributes non-universally
to each of the gaugino masses. Taking these shifts into account
and the GUT scale threshold corrections^{4}^{4}4
We found some discrepancies in Eq. (10) in Ref. [17].
One is an overall sign of the second parenthetic term in the right-hand side of
Eq. (10) which comes from the finite corrections of heavy GUT particles.
Subsequent equations indicate that this is merely a typo.
The other is the term in their Eq. (10),
which we believe should be .
This discrepancy originates from their Eq. (7), where we believe the
in the matrix should also be . [17] on
gaugino masses,
we find that the values of the gaugino masses at are

(16) | |||||

(17) | |||||

(18) |

where is the supersymmetry mass scale from the singlet field -term in Eq. (3). For our subsequent numerical work that will culminate in Fig. 2 we use and . These numerical values change slightly with the superpartner masses, but the qualitative features of the results stay the same. Furthermore, as we shall emphasize, these quantities are unambiguously calculable given knowledge of the low-energy superpartner spectrum.

The overall scale of the gaugino masses cannot be predicted; however, there are some interesting correlations among ratios of the gauginos. It is convenient to define the quantities

(19) |

The ’s are defined at the unification scale , and are unambiguously measurable given knowledge of the superpartner spectrum at the low scale and of course the beta functions of the MSSM up to the scale. Uncertainties in the extracted ’s from measurements would spring from uncertainties in superpartner masses and couplings, uncertainties in the all-orders beta functions in the renormalization group evolution, and uncertainties in the measured gauge couplings, most especially .

The values of and extracted from measurement will have discriminating power in the GUT scale parameter space of minimal supersymmetric . In this sense, we are testing the theory. There are four parameters of the GUT theory that are affecting the ratios of the gaugino mass values at ,

(20) |

Fitting four parameters to the two observables does not sound particularly enlightening, but there are a few interesting observations one can make about the underlying GUT model and the values.

For example, in minimal there is a relationship between terms and terms that must be satisfied in order to solve the doublet-triplet splitting problem,

(21) |

One solution to realize this relationship is the hypothesis of universal -terms and -terms . Under this hypothesis, possible regions of and are shown in Fig. 2 with and assuming and . As one can see from Fig. 2, a relative sign between and tends toward negative in the case, and toward positive in the non-zero cases. Also the corrections can be larger as gets larger. Therefore, there is an interesting opportunity to unveil a crucial role of the non-zero effect if we achieve precise enough determinations of gaugino masses at .

The effect on gaugino masses is an important one. Without it, the corrections to the gaugino mass ratios at the high scale fall along the rather narrow band in Fig. 2. Non-zero means, for example, that both and can be large and negative which is otherwise impossible.

There are two interesting limits to consider to illustrate how patterns of fundamental parameters can alter expectations of gaugino masses. One limit is when and the only corrections to the gaugino masses come from corrections. In that case, both and are approximately , a negative but small mismatch of gaugino masses at the high scale. Measuring these parameters to the sub-percent level is challenging even for a linear collider. We will outline measurement prospects below. In any event, it would perhaps be easier to rule out than confirm it by experiment. Thus, if both ’s are positive or one has a magnitude much bigger than , we will know that minimal with negligible and terms is not supported by the data.

The other interesting limit that eliminates the effect on the gaugino mass ratios is when , but and are non-zero. Since the contribution always is prefactored by , the value has no effect on the gaugino mass ratios in this case. Thus, variations of over its full range yields a line going through the origin that connects the two multi-line intersections in Fig. 2. That range is characterized by

(22) |

Therefore, any deviations beyond 10% would be firm evidence against this scenario, and even effects that deviate from the line would disaffirm the hypothesis.

Finally, we comment on the prospects of measuring and to the precisions required to make interesting suppositions about minimal . Very precise measurements of all superpartner masses and couplings are crucial. Given precise measurements of these quantities at the low scale, the scale can be derived unambiguously. The two-loop evaluation of the gaugino masses up to this scale is a well-defined prescription [18]. Blair et al. [19] have shown that a high-energy linear collider is capable of measuring gaugino masses well enough at the low-scale that a measurements at even the percent level can be discerned. Measuring down to this accuracy is not as easy, but it appears possible that even few percent could be established given careful analysis of LHC and linear collider data. The studies of Ref. [19] are very encouraging in that we believe they show that a linear collider along with the LHC could make a significant impact on our ability to draw interesting distinctions between GUT scale theories.

In conclusion, we have seen that no analysis of GUT gauge coupling unification can be complete without taking into account NRO corrections to the gauge kinetic function, and the expected size of these corrections from naive dimensional analysis suggests that they can play a decisive role in whether or not a theory is even viable. This is the case for minimal supersymmetric , where the adjoint-Higgs NRO corrections can save the theory. Furthermore, we have shown that there are experimental consequences at the low scale, and have illustrated how careful measurements of the gaugino mass spectrum can discern ideas, such as whether minimal with the universal -term and -term is viable. Further theoretical and experimental ideas would then be required to more definitively establish the theory or falsify it.

## Acknowledgements

This work was supported in part by the Department of Energy and the Alfred P. Sloan Foundation.

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