## July 5, 2007

### Beyond the MSSM

One of the talks at Strings 2007 that I surely would have blogged about, but for my laptop’s ailments was Nati Seiberg’s report on his most recent paper with Dine and Thomas.

As has become painfully familiar, the MSSM has a wee bit of a problem. At tree level, the mass of the lightest Higgs is lighter than that of the $Z$, $m_h \lt m_Z \cos(2\beta)$ Radiative corrections, due to stop and top loops, push the mass up. But, to get a heavy-enough Higgs compatible with the LEP-II bounds, requires a very heavy stop. That, in turn, makes the MSSM rather finely-tuned.

There are lots of ways to get around this. But, surprisingly, no one had attempted a general effective field theory analysis. Imagine there’s new physics at some scale, $M$. What new operators does this induce in the effective Lagrangian? And how do they affect the Higgs mass?

More broadly, the MSSM predicts tree-level relations among the masses of the five Higgses:

(1)\begin{aligned} m^2_h &= \frac{1}{2} \left(m_Z^2 +m_A^2 - \sqrt{{(m_A^2-m_Z^2)}^2+4 m_A^2 m_Z^2 \sin^2(2\beta)}\right)\\ m^2_H &= \frac{1}{2} \left(m_Z^2 +m_A^2 + \sqrt{{(m_A^2-m_Z^2)}^2+4 m_A^2 m_Z^2 \sin^2(2\beta)}\right)\\ m^2_{H^\pm} &= m_A^2 +m_W^2 \end{aligned}

where \begin{aligned} |\langle H_u\rangle | &= v \sin\beta\\ |\langle H_d\rangle | &= v \cos\beta \end{aligned}

These relations are modified by (known) radiative corrections and by new operators in the effective Lagrangian, induced by new physics at some scale $M$.

Surprisingly, at order $(1/M)$, there are only two new operators. There’s a supersymmetry-preserving quartic term in the superpotential, $W_{\text{Higgs}} = \int d^2\theta\, \mu H_u H_d + \frac{\lambda_1}{M} {(H_u H_d)}^2$ and there’s a supersymmetry-breaking correction to the scalar potential, which can be written as $\int d^2\theta \mathcal{Z} \frac{\lambda_2}{M} {(H_u H_d)}^2$ where $\mathcal{Z}= \theta^2 m_{\text{SUSY}}$ is a spurion field. Corrections to the Kähler potential can, to this order, be absorbed in field redefinitions.

These two terms yield corrections to the quartic term in the Higgs scalar potential

(2)$\delta V = 2\epsilon_1 H_u H_d (H_u^\dagger H_u + H_d^\dagger H_d)+\epsilon_2 {(H_u H_d)}^2 + \text{h.c.}$

where $\epsilon_1 = \frac{\overline{\mu}\lambda_1}{M},\quad \epsilon_2 = - \frac{m_{\text{SUSY}}\lambda_2}{M}$

It’s useful to study the problem in the limit of small $\eta \equiv \cot(\beta)$. This is the limit in which $h$ is predominantly $H_u$ and its tree-level mass in the MSSM is a large as can be. Seiberg et al hold the mass of the CP-odd state, $m_A$, fixed as they take the limit of large $\tan(\beta)$.

In this limit, we can expand (1) and include the corrections due to (2) \begin{aligned} m^2_h &= m_Z^2 -\frac{4 m_Z^2 m_A^2}{m_A^2-m_Z^2}\eta^2 + \frac{16 m_A^2}{m_A^2-m_Z^2} v^2 \eta Re(\epsilon_1) + \mathcal{O}(\eta^2\epsilon,\epsilon^2)\\ m^2_H &= m_A^2 +\frac{4 m_Z^2 m_A^2}{m_A^2-m_Z^2}\eta^2 +4 v^2 Re(\epsilon_2)- \frac{16 m_Z^2}{m_A^2-m_Z^2} v^2 \eta Re(\epsilon_1)+\mathcal{O}(\eta^2\epsilon,\epsilon^2)\\ m^2_{H^\pm} &= m_A^2 +m_W^2 +2 v^2 Re(\epsilon_2) \end{aligned}

If $\eta$ is small, but still larger than the $\epsilon_i$, these are the dominant corrections to the MSSM expressions for the Higgs masses. At $\mathcal{O}(\epsilon^2)$, we need to include the effect of further operators, suppressed by $1/M^2$. If we’re only interested in the contributions to $m_h^2$, there are a half-dozen of these, all corrections to the Kähler potential.

Even ignoring these higher corrections, it’s quite easy to reconcile the LEP II bounds with new physics in the 1-5 TeV range. In particular, a light top squark, or violations of the (loop-corrected) mass relations (1) — e.g. $m_h^2 +m_H^2 = m_Z^2 +m_A^2,\quad m_{H^\pm}^2 = m_W^2 + m_A^2$ which are $\tan(\beta)$-independent — will provide a direct window into new beyond-the-MSSM physics.

What’s really surprising about their result is that it is so simple (only two operators at leading order) and that it wasn’t found 25 years ago.

Posted by distler at July 5, 2007 11:36 AM

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### Re: Beyond the MSSM

“But, surprisingly, no one had attempted a general effective field theory analysis. Imagine there’s new physics at some scale, M. What new operators does this induce in the effective Lagrangian?And how do they affect the Higgs mass?”

Except unsurprisingly, they had. e.g. see hep-ph/0301121 and related references.

(I have no stake in this myself, but correct assignment of credit is important)

Posted by: anon on July 6, 2007 3:26 AM | Permalink | Reply to this

### Re: Beyond the MSSM

Except unsurprisingly, they had. e.g. see hep-ph/0301121 and related references.

Brignole et al looks like an interesting paper (which is cited by Seiberg and collaborators, so they’re aware of it). It differs from the present analysis in a crucial respect: it introduces additional light degrees of freedom to the MSSM, in the form of an $SU(3)\times SU(2)\times U(1)$ singlet chiral superfield, $T$. (For very specific couplings of $T$, this model is called the NMSSM. But Brignole et al consider more general couplings of $T$, including higher effective-dimension couplings, suppressed by powers of $1/M$.)

In the limit where $T$ is very heavy, you can, indeed, integrate it out and obtain an effective field theory along the lines of Dine, Seiberg and Thomas. But, in Brignole et al’s analysis, $m^2_T\sim m_{\text{SUSY}}^4/M^2$.

So, you’re right. Seiberg and company are not the first analysis of this general sort. But theirs is (to my, admittedly imperfect, knowledge), the first without additional light degrees of freedom beyond the MSSM.

Posted by: Jacques Distler on July 6, 2007 8:15 AM | Permalink | PGP Sig | Reply to this

### Re: Beyond the MSSM

Thank you for your interest in our paper hep-ph/0301121
(BCEN), especially to the “anon” who raised the fact that the approach followed by Dine, Seiberg and Thomas (DST) was presented by us four years ago. J. Distler (JD) makes some comments about the differences between the two papers and we would like to comment back.

* First of all, JD says in his comment that BCEN

”.. differs from the present analysis [DST] in a crucial respect: it introduces additional light degrees of freedom to the MSSM, in the form of an SU(3)×SU(2)×U(1) singlet chiral superfield, T”.

Well, you need a superfield (sometimes called the “spurion”) to break SUSY and of course DST also introduce that field (they call it Z). The couplings of Z to H_u H_d in the DST paper are identical to the couplings of T in our paper. In particular the two leading operators of eqs.(15, 20) of DST are the two operators of our eq.(4.1) (second line). There are exactly the same number of fields (and independent constants) than in DST. The operators are identical. (In the notation of your first note in the blog, what you call lambda_1, lambda_2 is called l, l’ in BCEN.) In addition we gave the second order operators (both in the superpotential W and the Kahler potential K). Finally, you replace the VEV of the auxiliary component of Z (or T in BCEN) by m_susy(\tilde m in BCEN).

Then you can write the Higgs potential. The corresponding additional terms (eqs.16,21 in DST, or eq.2 of your note) are identical to the last line of eq.4.4 of BCEN, where lambda_5,6,7 are given by eqs.4.9 (keeping the leading term).

Incidentally, notice that this T (or Z) field has nothing to do with the singlet of the NMSSM, which couples to H_u H_d at the renormalizable level and is not responsible for SUSY breaking.

* Second, the fact that these operators can be very useful to solve the MSSM fine-tuning problem was already pointed out in BCEN. See e.g. sentences in the abstract and at the end of point 2 of the conclusions (“A further advantage of the extra quartic couplings…”). Actually we devoted a whole subsequent paper (hep-ph/0310137) to discuss this way-out to the MSSM fine-tuning problem. See the sentences in the abstract.

* The observation that these quartic operators can change dramatically the Higgs spectrum of the MSSM was also amply discussed and illustrated in BCEN. See point 3 of our conclusions: “The spectrum of the Higgs sector……”.

* The fact that all these effects require new physics beyond MSSM at M ~ few TeV was also remarked in our paper from the very title and first line of the abstract (and in many places of both papers: hep-ph/0301121 and hep-ph/0310137).

In summary, we are not saying that DST do not make anything new, but the points that have attracted your attention in your note (and some other) were really developed in full detail in our papers. We agree with “anon” that this should be properly acknowledged.

Alberto Casas

Posted by: Alberto Casas on July 10, 2007 12:20 AM | Permalink | Reply to this

### BCEN vs DST

Well, you need a superfield (sometimes called the “spurion”) to break SUSY and of course DST also introduce that field (they call it Z).

A spurion is, by definition, nondynamical. The only nonzero component of $Z$ is a constant $F$ component.

The fact that, by contrast, $T$ is dynamical is crucial to your analysis. See, for instance, your equation (3.12). And it leads to results that differ markedly from those of DST.

It seems clear that you see this as some dynamical model, rather than as a general effective Lagrangian analysis. In your “examples” (section 5 of your paper), you simply drop one or the other of the effective dimension-5 operators of DST.

[To further add to my confusion, you include some effective dimension-6 operators in those “examples”, but I don’t think you agree with the analysis of the effective dimension-6 operators of DST.]

There have been, over the years, many papers which have solved the “little hierarchy problem” by adding new degrees of freedom to the MSSM. In analysing those models, one (obviously) finds corrections of precisely the form found by DST in their analysis.

But, to my knowledge, DST are correct in asserting that theirs is the first general effective Lagrangian analysis of the problem.

Posted by: Jacques Distler on July 10, 2007 9:22 AM | Permalink | PGP Sig | Reply to this

### Re: BCEN vs DST

“A spurion is, by definition, nondynamical. The only nonzero component of Z is a constant F component. The fact that, by contrast, T is dynamical is crucial to your analysis. See, for instance, your equation (3.12).”

You can define a spurion as non-dynamical, it is a matter of definition, but it seems clear that if the breaking of SUSY is spontaneous (not explicit), the field responsible for the breaking is dynamical. Otherwise, where is the goldstino in the low-energy theory? Obviously, the goldstino is essentially the fermionic component of the “spurion” (and is dynamical, see e.g. Polchinski and Susskind, “Breaking of Supersymmetry at Intermediate Energy”).

If the mass of the field responsible for SUSY breaking is large, as it often happens, one can neglect all the dynamical effects of that field in the effective theory. Then it is sensible to use a (non-dynamical) spurion as a convenient way to parametrize the effects of SUSY breaking. This is also OK if the breaking is explicit (see Siegel, “Fields”). However there is nothing wrong in considering the possible effects of the dynamics of the field responsible for the breaking. Obviously, the effective theory in this case has more contributions (which are of second order). In other words the effective theory in the case of a non-dynamical spurion is a subset of the effective theory when the dynamics of the “spurion” is considered, and at the leading order they coincide. This is exactly what happens when comparing BCEN with DST.

One may think that it is not necessary to include in practice the dynamical effects of T. This is true in many cases. However, since in this framework the SUSY breaking scale and the M scale are not so distant, it makes sense to write these contributions as well.

“It seems clear that you see this as some dynamical model, rather than as a general effective Lagrangian analysis.”

We do a general effective Lagrangian analysis. This does not mean that we have to ignore that the heavy fields are dynamical and can mediate to produce effective interactions in the low-energy theory (as Weinberg-Salam and the Fermi theory). The fact that the T field is explicitly present in our equations 3.12 and 4.4 may have led you to think that our effective theory has in fact more degrees of freedom than the MSSM. However, one can integrate out the T field to get a potential which depends just on the usual MSSM degrees of freedom. And we did that too! The last part of sect.4.2 is devoted to this task (see “..one can integrate out the T scalars from the beginning..”). We give the explicit expressions of the quartic couplings in this case too. As mentioned above they coincide with DST at the leading order. Making the T-mass large, the coincidence is complete.

“There have been, over the years, many papers which have solved the “little hierarchy problem” by adding new degrees of freedom to the MSSM.”

Again, we do not add any new degree of freedom to the MSSM, only the field responsible for SUSY breaking, which must be there, although it may be integrated out if desired. We emphasize in the paper, and also in hep-ph/0310137 and hep-ph/0410298, that our proposal to alleviate the fine-tuning problem is to make the scale of SUSY breaking lower, so that the SUSY breaking contributions to the usual Higss (just Higgs) quartic couplings are sizeable (see also eq.30 of hep-ph/0410298 for a very explicit statement of this). The only use of T for this bussiness is to break SUSY at a quite low scale.

Finally, you say: “DST are correct in asserting that theirs is the first general effective Lagrangian analysis of the problem.”
Well, I do not think they say that. Actually they say that “An effective action analysis of the MSSM Higgs sector has also been considered in earlier work [BCEN]”

Again, I do not criticise the work of DST, which I find very interesting (nor Seiberg’s talk, which I agree was really brilliant), my only concern is that our “early work” be correctly considered.

Thanks for your attention, Alberto Casas

Posted by: Alberto Casas on July 12, 2007 9:25 AM | Permalink | Reply to this

### Re: BCEN vs DST

Otherwise, where is the goldstino in the low-energy theory?

In supergravity, the goldstino is absorbed as the longitudinal component of the gravitino. When you decouple gravity (as has been done both in your work and in DST), it should decouple along with the rest of the gravitational sector.

Posted by: Jacques Distler on July 12, 2007 9:59 AM | Permalink | PGP Sig | Reply to this

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