Running couplings and the scale Lambda

The classical Coulomb branch contains a dimensionless coupling and a scalar expectation value. Quantum mechanics introduces a new dimensionful scale even though the microscopic theory has no mass parameter. How can this happen without choosing a preferred vacuum?

The answer is dimensional transmutation: a coupling specified at a reference energy can be replaced by a scale that records the same information. We will calculate it for pure four-dimensional N=2 $SU(2)$ theory, then explain why it controls the accuracy of the preceding semiclassical calculations. The running does not by itself lift the Coulomb branch.

Count the fields that renormalize the coupling

Keep eight real supercharges, no matter hypermultiplets, and $t^a=\sigma_a/2$ with $\operatorname{Tr}_2(t^at^b)=\delta^{ab}/2$. The N=2 vector contains one gauge field, two left-handed Weyl fermions and one complex scalar, all in the adjoint. Its N=1 presentation makes four supercharges manifest; it does not remove either fermion.

Let $\mu$ be a positive renormalization scale. The one-loop gauge beta function is

$$\beta(g)\equiv\mu\frac{dg}{d\mu} =-\frac{g^3}{16\pi^2}b_0+O(g^5),\qquad b_0=\frac{11}{3}C_A-\frac23\sum_{\rm Weyl}T(R_f) -\frac13\sum_{\rm complex}T(R_s). \tag{SW04.1}$$

This formula includes the gauge-fixing and ghost contribution in $11C_A/3$. The representation index is defined by $\operatorname{tr}_R(T^a_RT^b_R)=T(R)\delta^{ab}$, and $C_A=T(\mathrm{adj})$. These are indices, not the number of generators. For $SU(2)$, $(T^a_{\rm adj})_{bc}=-i\epsilon_{abc}$ gives $\operatorname{tr}_{\rm adj}(T^a_{\rm adj}T^b_{\rm adj})=2\delta^{ab}$. Thus $C_A=2$ and

$$b_0=\left(\frac{11}{3}-2\frac23-\frac13\right)2 =\boxed{4}. \tag{SW04.2}$$

The scalar is already complex in (SW04.1); counting it twice would be an error. Counting “three adjoint fields” once more would also double-count the color trace. As a separate check, an N=1 vector contributes $3C_A$ and its adjoint chiral contributes $-C_A$, again giving $2C_A=4$. The input one-loop formula and its supersymmetric regrouping are Tachikawa, (3.1.1)–(3.1.6).

The negative beta function means that $g$ decreases as $\mu$ increases: the theory is asymptotically free. It does not mean that its long-distance coupling is small.

Integrate the logarithm and define the scale

It is simpler to evolve the inverse coupling:

$$\mu\frac{d}{d\mu}\frac{8\pi^2}{g^2}=b_0=4,\qquad \frac{8\pi^2}{g^2(\mu)} =\frac{8\pi^2}{g^2(\mu_0)}+4\log\frac{\mu}{\mu_0}. \tag{SW04.3}$$

For an arbitrary definition of a measured coupling, (SW04.3) states the leading weak-coupling logarithm. To give a precise holomorphic scale definition, use a supersymmetry-preserving Wilsonian convention: integrate modes above $\mu$ while retaining lower-energy fields, and put their local gauge-kinetic coefficient in

$$\tau_W(\mu)=\frac{\theta}{2\pi}+\frac{4\pi i}{g_W^2(\mu)}. \tag{SW04.4}$$

Here the subscript labels this coefficient, not another particle species. In a holomorphic convention it depends holomorphically on the complexified coupling. Perturbation theory does not depend on the theta angle, since that term is a total derivative in the perturbative sector. A higher-loop correction proportional to powers of $g_W^2$, hence inverse powers of $\operatorname{Im}\tau_W$, would not be holomorphic. The allowed perturbative shift is the one-loop logarithm:

$$\mu\frac{d\tau_W}{d\mu}=\frac{2i}{\pi}. \tag{SW04.5}$$

This is perturbative one-loop exactness in the stated convention. N=2 supersymmetry also relates the two-derivative scalar and vector normalizations, so one must not add an unrelated scalar wave-function coupling. It does not follow that every coupling defined by fitting an arbitrary scattering observable is literally (SW04.4): finite matching, momentum dependence and higher-derivative effects must still be specified. A one-particle-irreducible action integrates the retained light modes too, and can contain nonlocal momentum dependence. Tachikawa, (3.1.7)–(3.1.11) and the following N=2 discussion.

Define the complex scale through its fourth power:

$$\boxed{\Lambda_h^4=\mu^4e^{\,2\pi i\tau_W(\mu)} =\mu^4\exp\!\left[-\frac{8\pi^2}{g_W^2(\mu)}+i\theta\right].} \tag{SW04.6}$$

Its logarithmic derivative is $4+2\pi i(2i/\pi)=0$, so it is invariant under this perturbative flow. Its magnitude satisfies

$$|\Lambda_h|=\mu_0\exp\!\left[-\frac{2\pi^2}{g_W^2(\mu_0)}\right], \qquad \frac{8\pi^2}{g_W^2(\mu)}=4\log\frac{\mu}{|\Lambda_h|}. \tag{SW04.7}$$

The fourth power contains the theta angle without choosing a fourth-root branch. A particular $\Lambda_h$ needs such a choice; $|\Lambda_h|$ does not. The subscript $h$ records the holomorphic renormalization convention. The scale $\Lambda$ that will normalize the Seiberg–Witten curve must be matched to this convention by a finite constant, not identified from the beta coefficient alone.

Dimensional transmutation has exchanged the pair $(\mu_0,g_W(\mu_0))$, with its redundant choice of reference point, for $|\Lambda_h|$. It has not produced a new adjustable mass term. The apparent divergence of (SW04.7) at $\mu=|\Lambda_h|$ says that this perturbative expression has left its weak-coupling domain; it is not a calculation of the actual strong-coupling physics.

Worked example: a large hierarchy from one coupling

In a specified holomorphic scheme take $g_W(\mu_0)=1$. Then

$$\frac{|\Lambda_h|}{\mu_0}=e^{-2\pi^2} \simeq2.67529\times10^{-9}.$$

Running down by a factor of one hundred, while all the non-Abelian fields are retained, gives

$$\frac{8\pi^2}{g_W^2(\mu_0/100)} =8\pi^2-4\log100,\qquad g_W(\mu_0/100)\simeq1.14206. \tag{SW04.8}$$

The coupling has grown, while the reference-independent scale computed from either endpoint is unchanged. A small change in $1/g_W^2(\mu_0)$ changes the hierarchy exponentially; this is compatible with logarithmic running.

Stop the non-Abelian running at the charged threshold

Three scales now play different roles:

• $\mu$ is the renormalization point used to describe the same theory.
• $M_W\simeq\sqrt2|a|$ is a charged-particle threshold in a chosen weak-coupling vacuum.
• $E$ is the momentum scale of the experiment in that vacuum.

The coordinate $a$ keeps the first Seiberg–Witten normalization, $a\sim\sqrt{2u}$. It is not a canonically normalized scalar expectation value at every scale. At tree level the preceding lesson wrote the canonical scalar as $\phi=b\,t^3$ and gave $a_{\rm cl}=gb$.

If $M_W\gg|\Lambda_h|$, run the non-Abelian theory down to a matching point of order $M_W$, then remove the charged multiplet for processes with $E\ll M_W$. The remaining N=2 $U(1)$ multiplet is neutral. It does not continue the same charged-particle logarithm with coefficient four down to $E$. Its local two-derivative coefficient instead depends on the vacuum through the threshold that was removed.

Holomorphy turns the magnitude logarithm into a local complex logarithm. On a weak-coupling patch, choose a continuous branch and write

$$\tau_{\rm eff}(a)=\frac{2i}{\pi}\log\frac{a}{\Lambda_h} +\kappa+O\!\left(\frac{\Lambda_h^4}{a^4}\right),\qquad \operatorname{Im}\tau_{\rm eff} =\frac2\pi\log\frac{|a|}{|\Lambda_h|} +\operatorname{Im}\kappa+\cdots . \tag{SW04.9}$$

Here $\kappa$ is a finite complex matching constant. The factor $\sqrt2$ between $M_W$ and $|a|$ contributes to it. The logarithmic slope is fixed by $b_0$; its additive constant is not. Controlled use needs both $|a|/|\Lambda_h|\gg1$ and $E/M_W\ll1$. Positivity of $\operatorname{Im}\tau_{\rm eff}$ is necessary for the electric kinetic term but, by itself, is not a quantitative weak-coupling criterion.

There is a source-normalization trap here. Tachikawa, (4.1.4)–(4.1.8) defines his Abelian coupling so that $\tau_{U(1),T}=2\tau_{SU(2),T}$ at tree level. His Abelian logarithm therefore has twice our slope. In the first-paper period convention used here,

$$a_T=\frac{a}{\sqrt2},\qquad a_{D,T}=\sqrt2\,a_D,\qquad n_T=2n_e,\quad m_T=n_m,\qquad \tau_{U(1),T}=2\tau_{\rm eff}. \tag{SW04.10}$$

The derivative relation and mass formula transform together. The coordinate rescaling changes a finite logarithmic constant; it does not change $b_0$ of the microscopic $SU(2)$ theory.

A finite constant is not another loop coefficient

The first paper writes its perturbative prepotential as $\mathcal F_{\rm pert}=\frac{i}{2\pi}a^2\log(a^2/\Lambda_F^2)$. We call the scale in this displayed expression $\Lambda_F$ to keep track of its finite convention. Using the already introduced relation $\tau_{\rm eff}=\mathcal F''$ gives

$$\tau_{\rm eff,pert} =\frac{i}{\pi}\left[\log\frac{a^2}{\Lambda_F^2}+3\right]. \tag{SW04.11}$$

Both derivatives matter: differentiating the logarithm produces the $+3$. It can be absorbed into a zero-constant logarithm by defining $\Lambda_{\log}=e^{-3/2}\Lambda_F$. No physical coupling changes when this replacement is made consistently. Neither notation fixes the absolute matching to the later curve by itself. Seiberg–Witten, (2.15) and the opening of section 3.

More generally, a finite redefinition $\tau_W'=\tau_W+\delta$ implies $\Lambda_h'^4=e^{2\pi i\delta}\Lambda_h^4$. Keeping (SW04.9) unchanged requires $\kappa'=\kappa-\delta$ on compatible branches. The beta coefficient, dimensional transmutation and physical predictions survive this change. A quoted numerical value of a dynamical scale requires its convention. Tachikawa makes this qualification explicitly in footnote 5.

Check the scale against the R anomaly

The classical $U(1)_r$ assigns charge two to $\phi$ and charge one to each adjoint Weyl fermion. In particular $a$ has charge two and $u$ has charge four. This is the first paper’s $U(1)_R$, not an arbitrary N=1 R-charge assignment.

In Euclidean spacetime define the instanton integer using the rescaled connection $\mathcal A=gA$:

$$k=\frac1{16\pi^2}\int d^4x\, \operatorname{Tr}_2(\mathcal F_{\mu\nu}\widetilde{\mathcal F}_{\mu\nu}), \qquad e^{-S_k}=e^{-8\pi^2 k/g_W^2+ik\theta} \quad(k>0). \tag{SW04.12}$$

Choose the orientation in which each left-handed adjoint Weyl operator has index $2T(\mathrm{adj})k=4k$. The two fermions therefore supply $8k$ Grassmann zero-mode coordinates. For an active rotation $\eta_j'=e^{i\alpha}\eta_j$, Berezin integration gives $d\eta_j'=e^{-i\alpha}d\eta_j$. Their measure acquires $e^{-8ik\alpha}$.

At fixed theta this is invariant for every integer $k$ only if $\alpha=\pi\ell/4$: the continuous symmetry is reduced to $\mathbb Z_8$. A formal compensating shift $\theta\mapsto\theta+8\alpha$ cancels the Jacobian. Such a transformation of a coupling is a spurion rule relating parameter values, not a continuous symmetry with all couplings held fixed. Equivalently $\Lambda_h^4$ has formal R charge eight. The mode-index input and theta weight are Tachikawa, (3.2.1)–(3.2.11); the residual group is Seiberg–Witten, section 2.2.

This provides two checks. First, $\Lambda_h^4/a^4$ is dimensionless and neutral under the spurion rule, precisely the small parameter allowed in (SW04.9). Its powers organize instanton corrections; symmetry alone does not calculate their coefficients. Second, the generator $\alpha=\pi/4$ sends $u$ to $-u$. At a generic nonzero vacuum only the $\mathbb Z_4$ subgroup fixes $u$. An anomaly of the action’s continuous symmetry and spontaneous breaking of its surviving discrete subgroup are different statements.

The first paper’s (2.16) distinguishes perturbative one-loop exactness from these nonperturbative corrections. Neither statement is permission to extend the weak-coupling logarithm through the strongly coupled region.

Exercises

1. Change the theory before using its beta function

Compare three $SU(2)$ theories: a gauge field alone; an N=1 vector alone; and the pure N=2 vector used here. Compute $b_0$ in each. Then add one full adjoint hypermultiplet to the N=2 theory and recompute the one-loop coefficient. What can the coefficient alone establish?

Hint 1

An N=1 vector has one Weyl fermion. A full adjoint hypermultiplet has two Weyl fermions and two complex scalars.

Hint 2

Each adjoint has index two. A zero value cancels a one-loop logarithm; distinguish that calculation from a general theorem about all quantum effects.

Solution

The three coefficients are $22/3$, $22/3-4/3=6$, and $22/3-8/3-2/3=4$. Thus “pure” must be accompanied by the amount of supersymmetry; it does not specify the same fields in all three cases.

The adjoint hypermultiplet contributes $-2(2/3)(2)-2(1/3)(2)=-4$, leaving $b_0=0$. It has twice the N=1 chiral content; adding only one chiral would leave $b_0=2$ instead. With the required N=2 interactions and zero hypermultiplet mass the full theory is N=4 Yang–Mills. Its stronger conformal properties use additional symmetry information, not just the arithmetic that its one-loop coefficient vanishes. In a general theory, a vanishing one-loop beta coefficient alone is insufficient to establish an exact fixed point.

For the distinction between a vanishing leading coefficient and a fixed point, see fixed points and scaling.

2. Move a finite constant without changing a prediction

Suppose one convention gives $\tau_{\rm eff}=(2i/\pi)\log(a/\Lambda_h)+\kappa$ on a fixed patch. Another defines $\Lambda_h'=2\Lambda_h$. Find $\kappa'$ so the same $\tau_{\rm eff}$ is obtained. Separately verify the factor relating $\Lambda_{\log}$ and $\Lambda_F$ in (SW04.11). Why is setting every scale called Lambda equal not an innocent abbreviation?

Hint 1

Use $\log(a/(2\Lambda_h))=\log(a/\Lambda_h)-\log2$ on the chosen branch.

Hint 2

In the second comparison, $\log(a^2/\Lambda_{\log}^2)$ must reproduce $\log(a^2/\Lambda_F^2)+3$.

Solution

The required constant is $\kappa'=\kappa+(2i/\pi)\log2$. The logarithm lost that imaginary constant, and the matching coefficient restores it. The microscopic definition would correspond to $\delta=-(2i/\pi)\log2$, consistent with $\Lambda_h'^4=e^{2\pi i\delta}\Lambda_h^4=16\Lambda_h^4$ and $\kappa'=\kappa-\delta$.

The other equality gives $\Lambda_{\log}^2=e^{-3}\Lambda_F^2$, or $\Lambda_{\log}=e^{-3/2}\Lambda_F$ with compatible phases. Identifying scales while discarding their compensating constants changes the predicted inverse coupling. The logarithmic slope is universal at this order; an absolute finite scale normalization still needs a matching prescription.

3. Separate a weak vacuum from a low-energy experiment

Take a weak vacuum with $|a|/|\Lambda_h|=10^4$ and choose the illustrative matching convention $\kappa=0$, with $a/\Lambda_h$ positive real. Compute the leading $g_{\rm eff}^2=4\pi/\operatorname{Im}\tau_{\rm eff}$ and the nominal instanton parameter. Compare experiments at $E=M_W/100$ and $E=2M_W$. Should the first experiment use $b_0=4$ to run the photon-only theory from $M_W$ down to $E$? Finally, what does the exact surviving R generator do to this vacuum’s $u$?

Hint 1

The ratio $|a|/|\Lambda_h|$ controls the coupling expansion. The separate ratio $E/M_W$ controls whether the charged fields may be removed.

Hint 2

The photon-only N=2 multiplet has no electrically charged component. For the R transformation use charge four for $u$ and $\alpha=\pi/4$.

Solution

At the displayed order,

$$\operatorname{Im}\tau_{\rm eff}=\frac2\pi\log10^4\simeq5.86348, \qquad g_{\rm eff}^2=\frac{2\pi^2}{\log10^4}\simeq2.14316, \qquad \left|\frac{\Lambda_h}{a}\right|^4=10^{-16}.$$

The loop expansion parameter $g_{\rm eff}^2/(16\pi^2)\simeq0.01357$ is small. The instanton ratio is also small; it does not specify the unknown coefficient multiplying a particular correction.

At $E=M_W/100$, a local derivative expansion can remove the charged multiplet. Its threshold has already entered $\tau_{\rm eff}(a)$; continuing its non-Abelian coefficient four below the threshold would count fields no longer present. At $E=2M_W$, charged production is possible, so the photon-only description is inadequate even though the vacuum is weakly coupled.

The surviving R generator sends $u\mapsto e^{4i(\pi/4)}u=-u$. Since this vacuum has $u\ne0$, it is a different vacuum related by a symmetry, not a gauge copy. The full discrete symmetry is a symmetry of the theory, while the particular vacuum preserves its subgroup fixing $u$.

We can now identify a weakly coupled vacuum, estimate its charged threshold and track finite normalization choices. Next, derive how a prepotential controls the couplings: N=2 supersymmetry relates the local scalar metric and photon action, while the derivative expansion sets their scope.

Paper guides

Connect this lesson to the original papers and their notation.

• Reading the first Seiberg–Witten paper