## October 15, 2007

### Gluino Masses

The MSSM is much-maligned for having many more parameters than the Standard Model. Of course, in the supersymmetric limit, it has no more parameters than the Standard Model. To the contrary, the Higgs quartic coupling is related to the gauge coupling, a simplification that is the source of a certain amount of trouble.

Supersymmetry breaking introduces a plethora of soft parameters. But, as we mentioned last time, we already have some quite stringent constraints on these parameters. And these have nontrivial implications for higher-energy physics.

But we’d like to do better. We’d like to extract some robust (that is to say, relatively model-independent) predictions for these soft parameters. The most promising place to look is at the gaugino masses, where Nilles and Choi have done a very nice analysis.

Both the gauge coupling and the gaugino masses arise from the holomorphic gauge coupling functions, $f_a$, in the supergravity action. The ratio $\tfrac{M_a(\mu)}{g^2_a(\mu)}$ is invariant under 1-loop renormalization group running. But it can receive important threshold corrections, in gauge-mediated supersymmetry breaking, from integrating out the heavy messenger fields, $\Phi$, which are charged under the Standard Model gauge group.

Writing the Wilsonian effective action at the cutoff scale, $\Lambda$

$\int d^4\theta C C^* \left(-3 e^{-K/3}\right) +\left[\int d^2\theta \left(\tfrac{1}{4} f_a W^{a\alpha}W^a_\alpha + C^3 W\right)+ h.c.\right]$

where $C$ is the chiral compensator, and the Kähler potential, $K$, can be expanded as

$K= K_0(X_I,X^*_I) + \sum_\Phi Z_\Phi(X_I,X^*_I) \Phi \Phi^* + \sum_m Z_m(X_I,X^*_I) Q_m Q^*_m$

where the $X_I$ are Standard Model singlet fields, with potentially non-vanishing F-components, $F_I$. $\Phi$ are messenger fields (if present), and the $Q_m$ are the (Standard Model) fields, which are light at the TeV scale.

One wants to run the gauge coupling functions down from $\Lambda$ to a TeV. Above the threshold where the $\Phi$ get a mass, the running is simply expressed in terms of

$\mathcal{F}_a(\mu) = Re\left(f_a^{(0)}\right) - \tfrac{1}{16\pi^2} \left(3 C_a -\sum_i C_a^i\right) \log\left(\tfrac{C C^* \Lambda^2}{\mu^2}\right) - \tfrac{1}{8\pi^2}\sum_i C_a^i \log\left(e^{K_0/3}Z_i\right) +\tfrac{1}{8\pi^2} \Omega_a$

The running couplings and gaugino masses are

\begin{aligned} \frac{1}{g_a^2(\mu)} &= \mathcal{F}_a(\mu)|_{C=e^{K_0/6}}\\ M_a(\mu) &= (F_I \partial_{X_I} + F_C\partial_C) \mathcal{F}_a(\mu)|_{C=e^{K_0/6}} \end{aligned}

Here, $f_a^{(0)}$ are the tree-level gauge coupling functions. The index $i$ runs over the matter fields, $Q_m$, and the messengers, $\Phi$. $C_a^i$ is the quadratic Casimir in representation $i$, and $C_a$ is the quadratic Casimir in the adjoint. $\Omega_a$ are the threshold corrections due to integrating out massive fields at the cutoff scale $\Lambda$.

Running down to the messenger scale, and integrating out the messengers, introduces threshold corrections. These produce shifts in the value of $M_a/g_a^2$. Let the mass terms for the messengers be

$\int d^2\theta C^3 X_\Phi \Phi \Phi^c + h.c.$

where $\langle X_\Phi\rangle = M_\Phi + \theta^2 F_{X_\Phi}$. Then the net effect of running from the cutoff scale down to a TeV can be written as

(1)\begin{aligned} \left.\frac{M_a}{g_a^2}\right|_{\text{TeV}} &=\tilde{M}_a^{(0)} + \tilde{M}_a^{(1)}|_{\text{anom}}+ \tilde{M}_a^{(1)}|_{\text{gauge}}+ \tilde{M}_a^{(1)}|_{\text{string}}\\ \tilde{M}_a^{(0)} &= \tfrac{1}{2} F_I \partial_{X_I} f_a^{(0)}\\ \tilde{M}_a^{(1)}|_{\text{anom}} &= \tilde{M}_a^{(1)}|_{\text{conformal}} +\tilde{M}_a^{(1)}|_{\text{Konishi}}\\ &= \frac{b_a}{16\pi^2}\frac{F_C}{C} - \frac{1}{8\pi^2}\sum_m C^m_a F_I \partial_{X_I} \log\left(e^{-K_0/3} Z_m\right)\\ \tilde{M}_a^{(1)}|_{\text{gauge}} &= - \frac{1}{8\pi^2} \sum_\Phi C^\Phi_a \frac{F_{X_\phi}}{M_\Phi}\\ \tilde{M}_a^{(1)}|_{\text{string}} &= - \frac{1}{8\pi^2} F_I \partial_{X_I} \Omega_a \end{aligned}

Here, $b_a = - (3C_a - \sum_m C^m_a)$ are the 1-loop $\beta$-function coefficients at a TeV,

$\frac{F_C}{C} = m_{3/2}^* + \tfrac{1}{3} F_I \partial_{X_I} K_0$

and the effect of the aforementioned threshold corrections at the messenger scale are subsumed in $\tilde{M}_a^{(1)}|_{\text{gauge}}$.

In any particular string compactification scenario, one or another of the terms in (1) will dominate. At the TeV scale, the ratios of gauge couplings are, approximately

$g^2_1\, :\, g^2_2 :\, g^2_3 = 1\, :\, 2\, :\, 6$

and, putting this together with (1), one obtains characteristic predictions for the ratios of gaugino masses.

In heterotic string compactifications (either weakly-coupled, or Hořava-Witten theory), the tree-level gauge coupling functions are universal (independent of $a$), and supersymmetry-breaking (to the extent that we have a satisfactory theory of moduli stabilization) is gravity or moduli mediated. Hence, the gaugino masses fall into the same pattern:

(2)$M_1\, :\, M_2 :\, M_3 = 1\, :\, 2\, :\, 6$

The same result obtains in IIB flux compactifications, with flux-induced SUSY-breaking (modulo some, possibly dubious assumptions about the stabilization of the Kähler moduli).

In gauge-mediated SUSY-breaking, it is the threshold corrections, $\tilde{M}_a^{(1)}|_{\text{gauge}}$, at the messenger scale, that dominate (1). However, if the $\Phi$ form complete $SU(5)$ multiplets (so that gauge coupling unification is preserved), then (2) still obtains.

Even in the “large-volume” model of Balasubramanian et al, where there is no gauge coupling unification, this same pattern emerges.

A different distinctive pattern emerges in anomaly mediation, where the $\tilde{M}_a^{(1)}|_{\text{conformal}}$ dominates (1). Using $b_a= (33/5,1,-3)$, one obtains the ratio of gaugino masses

(3)$M_1\, :\, M_2 :\, M_3 = 3.3\, :\, 1\, :\, 9$

Of course, pure anomaly mediation is problematic, as it leads to tachyonic slepton masses.

More common, at least in its stringy realizations, is a hybrid of (2) and (3), which Choi and Nilles dub the Mirage pattern

(4)$M_1\, :\, M_2 :\, M_3 = (1+ .66\alpha)\, :\, (2+0.2\alpha)\, :\, (6-1.8\alpha)$

where $\alpha$ is a constant of order 1. In the original KKLT model, with SUSY-breaking by a $\overline{D3}$ down a warped throat, one finds (4), with $\alpha=1$.

Perhaps the most important class of string compactifications which haven’t (yet) been fit into this framework were recently proposed by Acharya et al. These involve M-theory on $G_2$ manifolds, where the moduli are stabilized by nonperturbative effects.

Understanding those models (and extracting their characteristic prediction for the ratios of gaugino masses) is clearly a priority. I hope to return to them in a future post.

Posted by distler at October 15, 2007 11:10 PM

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### Re: Gluino Masses

But in the supersymmetric limit, we have a lot of masses -even the ones of the standard model- still zero, do we? Or are you telling that we are first breaking electroweak and only then, in a second separate step, breaking susy?

Posted by: A Rivero on October 16, 2007 1:14 PM | Permalink | Reply to this

### Re: Gluino Masses

1. In the supersymmetric limit, electroweak gauge symmetry is unbroken.
2. Nonetheless, the Yukawa couplings (which give rise to the masses of quarks and leptons, once symmetry-breaking does occur) are parameters in the supersymmetric Lagrangian.
Posted by: Jacques Distler on October 16, 2007 1:54 PM | Permalink | PGP Sig | Reply to this

### Re: Gluino Masses

And, if we are going to admit that, thanks to the Yukawa couplings, the mass values already exist “in potentia” in the supersymmetric limit, then… should we admit that all the rest, soft parameters included, exist too?.

It seems Aristotle is playing with us here.

Posted by: Alejandro Rivero on October 16, 2007 3:07 PM | Permalink | Reply to this

### Re: Gluino Masses

I’m sorry. You have completely lost me.

• I can count renormalizable couplings in the supersymmetric Lagrangian.
• I can count soft supersymmetry-violating operators.
• There are many more of the latter, than there are of the former.

I don’t know what Aristotle has to do with any of this.

Posted by: Jacques Distler on October 16, 2007 4:32 PM | Permalink | PGP Sig | Reply to this

### Re: Gluino Masses

I apologize. I let my head go after the initial comment “in the supersymmetric limit, it has no more parameters than the Standard Model” and started to think about how and when the rest of parameters of a supersymmetric model do appear, and then I let myself raptured into an aristotelian disgression about “potentia” and “actus”, pretty unphysical :-(

Posted by: Alejandro Rivero on October 17, 2007 1:34 PM | Permalink | Reply to this
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### Re: Gluino Masses

“Perhaps the most important class of string compactifications which haven’t (yet) been fit into this framework were recently proposed by Acharya et al. These involve M-theory on G 2 manifolds, where the moduli are stabilized by nonperturbative effects.

Understanding those models (and extracting their characteristic prediction for the ratios of gaugino masses) is clearly a priority. I hope to return to them in a future post.”

Hi Jaques,
I’ve been trying to understand these models too. What do you think about this paper which just came out?
http://arxiv.org/abs/0810.3285
Did they address the problems with gaugino masses?

Posted by: Mark on October 21, 2008 10:55 AM | Permalink | Reply to this

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