Soft Breaking, Spurions, and Controlled Decoupling

Soft supersymmetry breaking is a controlled way to deform a supersymmetric Lagrangian explicitly: the allowed dimensionful operators preserve the improved ultraviolet behavior of scalar masses, while splitting supermultiplets and shifting vacua. “Soft” is a renormalization statement, not a synonym for small, harmless, flavor blind, or continuously connected at arbitrarily large coefficient. This page derives the standard operators from spurions, checks a complete mass spectrum, follows their running and threshold matching, and distinguishes the supersymmetry-restoring limit from a large-soft decoupling path.

Required background. Use Holomorphic Couplings as Background Superfields for the spurion construction and Gauge–Matter Systems, F- and D-Term Potentials for the undeformed component theory.

Helpful background. A dynamical origin for the frozen spurion is described by Nonlinear Goldstino Dynamics and Constrained Effective Theory. Use Decoupling Theorems and Threshold Corrections when a soft mass crosses the matching scale.

Soft terms as explicit relevant deformations

Start with a renormalizable rigid $\mathcal N=1$ gauge theory whose superpotential is

$$W=\frac12\mu^{ij}\Phi_i\Phi_j +\frac16y^{ijk}\Phi_i\Phi_j\Phi_k.$$

With canonical kinetic terms, define the sign convention

$$\begin{aligned} -\mathcal L_{\rm soft}={}& (m^2)^i{}_j\,\phi_i^*\phi^j +\frac12\left(M_a\lambda^a\lambda^a+\text{h.c.}\right)\\ &+\left[ \frac16a^{ijk}\phi_i\phi_j\phi_k +\frac12b^{ij}\phi_i\phi_j +t^i\phi_i+\text{h.c.} \right]. \end{aligned}$$

Gauge invariance restricts every tensor; in particular $t^i$ is possible only for a gauge singlet. The parameters have dimensions

$$[M_a]=[a^{ijk}]=1, \qquad [(m^2)^i{}_j]=[b^{ij}]=2, \qquad [t^i]=3.$$

These are the standard holomorphic soft terms. Under the assumptions of a renormalizable theory and no dangerous gauge singlets, they do not reintroduce quadratic divergences into scalar masses. Logarithmic running remains, and vacuum energy or singlet tadpoles can still be power sensitive. Nonholomorphic cubic terms can be soft in restricted field contents, but singlets and mixed operators require a separate divergence analysis; engineering dimension alone is not a classification theorem. The all-orders power-counting origin of the standard list is due to Girardello and Grisaru 1982, pp. 65–76; component conventions and the practical classification are reviewed in Weinberg 2000, §27.7, pp. 155–157 and Martin 2016, §§5–6.

Several negative statements are as important as the definition:

• a soft parameter can be much larger than the masses one wishes to protect;
• arbitrary flavor matrices $(m^2)^i{}_j$ can generate flavor violation;
• complex $M,a,b$ can introduce new CP phases;
• negative scalar eigenvalues can destabilize the vacuum;
• a large soft coefficient can exceed the EFT cutoff;
• frozen soft terms break supersymmetry explicitly and do not require a massless goldstino.

Spurion derivation

Let a gauge-singlet background chiral superfield be

$$S=s+\sqrt2\theta\chi_S+\theta^2F_S.$$

Treating $S$ as a transforming background preserves superspace selection rules. Freezing $S=\theta^2F_S$ then generates explicit breaking in the visible theory. Representative operators are

$$\begin{aligned} &\int\!\mathrm d^2\theta\, \frac14\left(\frac{1}{g_a^2}+2c_a\frac{S}{M}\right) W^{a\alpha}W^a_\alpha &&\Longrightarrow&& M_a\sim c_a\frac{F_S}{M},\\ &\int\!\mathrm d^4\theta\, c^i{}_j\frac{S^\dagger S}{M^2} \Phi_i^\dagger e^V\Phi^j &&\Longrightarrow&& (m^2)^i{}_j\sim c^i{}_j\frac{|F_S|^2}{M^2},\\ &\int\!\mathrm d^2\theta\, \frac16c_A^{ijk}\frac{S}{M}\Phi_i\Phi_j\Phi_k &&\Longrightarrow&& a^{ijk}\sim c_A^{ijk}\frac{F_S}{M},\\ &\int\!\mathrm d^2\theta\, \frac12\widehat c_B^{ij}\frac{S}{M}\Phi_i\Phi_j &&\Longrightarrow&& b^{ij}\sim \widehat c_B^{ij}\frac{F_S}{M}. \end{aligned}$$

Here $\widehat c_B^{ij}$ has mass dimension one and may be proportional to the supersymmetric mass matrix $\mu^{ij}$. Numerical factors depend on gauge-kinetic and component normalizations; the scaling and symmetry selection rules do not. A Kähler operator linear in $S^\dagger$ can also generate supersymmetric masses or $b$ terms after auxiliary elimination, so a complete matching calculation must include field redefinitions and all operators at the same order.

If $S$ is the low-energy remnant of a dynamical hidden sector, the full theory may break supersymmetry spontaneously and contain a goldstino. The visible theory obtained after freezing and integrating out that sector is explicitly broken. Calling the frozen parameter an “F-term” does not by itself restore a conserved visible-sector supercurrent.

A complete mass-spectrum benchmark

Take two chiral multiplets $\Phi_\pm$ with

$$W=m\Phi_+\Phi_-$$

and add a common soft scalar mass and bilinear:

$$V_{\rm soft} =m_s^2\left(|\phi_+|^2+|\phi_-|^2\right) +\left(b\phi_+\phi_-+\text{h.c.}\right).$$

The two Weyl fermions form a Dirac fermion with mass squared $|m|^2$. In the scalar basis $(\phi_+,\phi_-^*)$, the Hermitian mass-squared matrix is

$$\mathcal M_B^2= \begin{pmatrix} |m|^2+m_s^2&b^*\\ b&|m|^2+m_s^2 \end{pmatrix},$$

with eigenvalues

$$m_{B,\pm}^2=|m|^2+m_s^2\pm|b|.$$

Each eigenvalue belongs to one complex scalar. The origin is stable exactly when

$$|m|^2+m_s^2>|b|.$$

As $m_s^2,b\to0$, both complex scalars become degenerate with the Dirac fermion and the supersymmetric spectrum is recovered. At fixed nonzero $m_s^2$ or $b$, the multiplet is split. If $|b|$ crosses $|m|^2+m_s^2$, a tachyon appears even though $b$ is a formally soft operator. Softness did not protect the vacuum.

The supertrace provides a check:

$$\operatorname{STr}\mathcal M^2 =2m_{B,+}^2+2m_{B,-}^2-4|m|^2 =4m_s^2.$$

Unlike an exactly supersymmetric spectrum, a softly broken one need not have zero supertrace after the soft masses are inserted.

Vacuum shifts for small soft parameters

Let $\varphi^I$ be real physical scalar coordinates after gauge fixing, and suppose the supersymmetric potential $V_0$ has an isolated minimum $\varphi_0$ with positive Hessian

$$(H_0)_{IJ}=\left.\frac{\partial^2V_0}{\partial\varphi^I\partial\varphi^J}\right|_{\varphi_0}.$$

For

$$V=V_0+\epsilon V_{\rm soft},$$

the implicit-function theorem gives

$$\delta\varphi^I =-\epsilon(H_0^{-1})^{IJ} \left.\frac{\partial V_{\rm soft}}{\partial\varphi^J}\right|_{\varphi_0} +O(\epsilon^2).$$

This is the controlled meaning of a “nearby soft deformation.” It requires an invertible physical Hessian and a shift small compared with the radius of convergence and any distance to a tachyon boundary. It fails on a moduli space, at a second-order transition, in an IR-singular infinite-volume limit, or when the selected vacuum jumps between disconnected basins.

If $V_0$ has flat directions, even an arbitrarily small soft term can select a point at order one in field space. One must first construct the effective potential on the moduli space rather than invert a zero Hessian.

Renormalization-group flow

Softness is stable under renormalization, but individual coefficients mix. In a convention with

$$16\pi^2\frac{\mathrm dg_a}{\mathrm d\log\mu}=b_ag_a^3,$$

the one-loop gaugino equation is

$$16\pi^2\frac{\mathrm dM_a}{\mathrm d\log\mu} =2b_ag_a^2M_a.$$

Therefore $M_a/g_a^2$ is one-loop invariant between thresholds. For a field $\Phi_i$, the scalar-mass beta function contains the characteristic terms

$$16\pi^2\frac{\mathrm dm_i^2}{\mathrm d\log\mu} =\text{Yukawa sums involving }m^2\text{ and }|a|^2 -8\sum_a g_a^2C_a(i)|M_a|^2 +\text{Abelian trace terms}.$$

For the simple ungauged superpotential $W=y\Phi_1\Phi_2\Phi_3$ with $a=Ay$, the Yukawa part is

$$16\pi^2\frac{\mathrm dm_1^2}{\mathrm d\log\mu} =2|y|^2\left(m_1^2+m_2^2+m_3^2+|A|^2\right),$$

and cyclically. These equations show why setting one soft term to zero at a high scale does not generally keep it zero: other soft coefficients regenerate it unless a symmetry forbids the mixing. The general one- and two-loop equations, with explicit index conventions, are given in Martin and Vaughn 1994, §§2–4.

Running must be combined with vacuum checks. A scalar eigenvalue positive at the matching scale can run negative and trigger symmetry breaking; conversely, a negative running mass parameter does not by itself determine the physical pole spectrum without minimizing the RG-improved potential.

Thresholds and controlled decoupling

Suppose a soft mass $M_{\rm soft}$ is much larger than an external scale $E$. To remove the heavy field:

1. run the full theory to a matching scale $\mu\sim M_{\rm soft}$;
2. match light-field amplitudes or 1PI functions, including finite threshold terms;
3. express the low-energy couplings in the EFT scheme;
4. run with the EFT beta functions;
5. include operators suppressed by powers of $E/M_{\rm soft}$.

In a mass-independent scheme, heavy fields do not disappear automatically from beta functions. Their logarithms are transferred into matching coefficients. Appelquist–Carazzone decoupling says that, under its renormalizable and scale-separation hypotheses, low-energy heavy-particle effects can be absorbed into renormalized parameters plus power-suppressed operators Appelquist and Carazzone 1975, pp. 2856–2861. It does not say that thresholds vanish or that the vacuum phase remains unchanged.

Two limits with different logical force

The restoration limit sends

$$M_a,a^{ijk},b^{ij},t^i,(m^2)^i{}_j\longrightarrow0$$

at fixed supersymmetric couplings and cutoff. For an isolated gapped vacuum and finite observables, perturbation theory gives a continuous return to supersymmetric Ward identities and degenerate multiplets. Flat directions, IR divergences, infinite volume, and phase boundaries can make the limit nonuniform.

The large-soft limit sends selected masses to infinity to remove superpartners. It is a different path. Along it,

• threshold corrections grow logarithmically or leave finite matching shifts;
• the light-field vacuum can change or disappear;
• a large coefficient can approach the mediation scale or cutoff;
• the RG flow may cross strong coupling or a phase transition;
• limits of infinite volume, continuum removal, and $M_{\rm soft}\to\infty$ need not commute.

Consequently, a soft path can define and sometimes compute a nearby nonsupersymmetric EFT. It cannot by itself prove that a remote nonsupersymmetric theory, such as pure Yang–Mills after all partners are removed, shares the same phase or nonperturbative observables. Such a conclusion requires an independently controlled interpolation with matching, no intervening transition, and a tracked observable.

Evidence table for a soft-deformation claim

Claim	Evidence required	What would falsify control
Operator is soft	Girardello–Grisaru class or explicit divergence analysis with singlets and symmetries stated	New quadratic scalar-mass divergence
Spurion origin	Superspace operator, mediation scale, dimensions, and auxiliary insertion	Missing operator at the same EFT order
Vacuum is nearby	Invertible physical Hessian and small computed $\delta\varphi$	Flat direction, tachyon crossing, or basin jump
RG prediction	Boundary conditions, scheme, beta functions, and thresholds	Running across a threshold without matching
Heavy partner decouples	$E\ll M$, matched parameters, and suppressed higher operators	Large logarithms or $M$ near the cutoff
SUSY is restored	All explicit breakings vanish and Ward identities recover uniformly	Residual spurion, anomaly, singular IR limit
Nonsupersymmetric phase is connected	Observable tracked along a path with no transition	Unchecked strong-coupling or phase boundary

This distinction between theorem-level softness, model-level stability, and path-level decoupling prevents one piece of evidence from being reused for a stronger claim.

Common pitfalls

“Soft means numerically small.” It does not. Softness concerns ultraviolet divergences; perturbative vacuum shifts require a separate small-parameter comparison.

Freezing a spurion but claiming spontaneous breaking. Once the source is nondynamical, the visible action is explicitly broken. Recovering a goldstino requires restoring and analyzing the dynamical sector.

Sending masses to infinity without matching. A mass-independent beta function still contains the heavy field until an EFT is matched. Dropping it by inspection misses logarithmic and finite thresholds.

Using a soft path as a phase theorem. Analyticity near zero soft breaking says nothing about an arbitrarily distant endpoint if a gap closes or a transition intervenes.

Exercises

1. Locate the tachyon boundary. Diagonalize the scalar mass matrix in the two-chiral benchmark and state the exact stability condition.

Solution

A phase rotation makes $b$ real and positive. The eigenvectors $(\phi_+\pm\phi_-^*)/\sqrt2$ have squared masses $|m|^2+m_s^2\pm|b|$. Both are positive exactly when $|m|^2+m_s^2>|b|$.

2. Derive the vacuum shift. Expand $\partial_IV(\varphi_0+\delta\varphi)=0$ through first order in $\epsilon$.

Solution

Since $\partial_IV_0(\varphi_0)=0$,

$$0=(H_0)_{IJ}\delta\varphi^J +\epsilon\,\partial_IV_{\rm soft}(\varphi_0)+O(\epsilon^2).$$

Multiplication by $H_0^{-1}$ gives the displayed formula. If $H_0$ has a zero mode, this step fails and the flat direction must be treated separately.

3. Verify the one-loop invariant. Show from the displayed beta functions that $\mathrm d(M_a/g_a^2)/\mathrm d\log\mu=0$ between thresholds.

Solution

Using the quotient rule,

$$\frac{\mathrm d}{\mathrm dt}\frac{M_a}{g_a^2} =\frac{1}{g_a^2}\frac{\mathrm dM_a}{\mathrm dt} -\frac{2M_a}{g_a^3}\frac{\mathrm dg_a}{\mathrm dt}.$$

Substituting $16\pi^2\dot M_a=2b_ag_a^2M_a$ and $16\pi^2\dot g_a=b_ag_a^3$ makes the two terms cancel.

4. Compare limits. In the mass benchmark, contrast $(m_s^2,b)\to0$ with $m_s^2\to\infty$ at fixed $m$.

Solution

The first limit restores the degenerate chiral multiplets and supersymmetric Ward identities, provided no IR singularity intervenes. The second removes the scalars but leaves the fermion; it requires matching at $m_s$, generates threshold corrections, and does not approach the supersymmetric theory. The two paths have different field content and logical meaning.

References

• Appelquist, T., and J. Carazzone. “Infrared Singularities and Massive Fields.” Physical Review D 11 (1975): 2856–2861. DOI.
• Girardello, L., and M. T. Grisaru. “Soft Breaking of Supersymmetry.” Nuclear Physics B 194 (1982): 65–76. DOI.
• Martin, S. P. “A Supersymmetry Primer.” Version 7, 2016. Open preprint.
• Martin, S. P., and M. T. Vaughn. “Two-Loop Renormalization Group Equations for Soft Supersymmetry-Breaking Couplings.” Physical Review D 50 (1994): 2282–2291; erratum 78 (2008): 039903. DOI. Open preprint.
• Weinberg, S. The Quantum Theory of Fields, Volume III: Supersymmetry. Cambridge: Cambridge University Press, 2000, §27.7. DOI.