Mass Tuning and Relevant Deformations

The previous page used the running quartic coupling and the running normalization of the thermal operator to derive the logarithmic specific-heat singularity at the four-dimensional upper critical dimension. That calculation had a quiet but essential assumption: the theory was close enough to criticality that the RG flow could run for many decades before being stopped by a physical mass.

This page makes that assumption explicit. A scalar theory near a continuous transition has at least one relevant parameter: the coefficient of $\phi^2$. This parameter is not just another coupling. It decides whether the infrared theory is critical or massive. If it is tuned to a special value, the correlation length diverges and the logarithmic critical behavior of the previous page is visible. If it is not tuned, the flow stops at the inverse correlation length, and the long-distance theory becomes insensitive to further critical running.

The main lessons are:

$$\text{bare mass} \ne \text{physical mass}, \qquad m=\xi^{-1}, \qquad \lambda_{\rm phys}=\lambda(k=m),$$

and criticality is the condition

$$m_{\rm phys}=0,$$

which usually requires an additive tuning of the bare quadratic coefficient.

Relevant directions and the critical surface

Cutoff action and stopping scale

We use a cutoff scalar theory in four Euclidean dimensions,

$$S_\Lambda[\phi]=\int d^4x\left[ {1\over2}(\partial_\mu\phi)^2+{1\over2}r_0\phi^2+{\lambda_0\over4!}\phi^4+\sum_i c_{i,0}\mathcal O_i \right],$$

where $r_0$ is the bare quadratic coefficient. The symbol $r_0$ is useful because in statistical mechanics it is proportional, after an additive shift, to the reduced temperature. In particle-physics notation one often writes $r_0=m_0^2$.

For the one-component theory,

$$a={3\over16\pi^2}, \qquad \lambda(k)={\lambda_0\over 1+a\lambda_0\log(\Lambda/k)},$$

and the leading-log normalization of a $\phi^2$ insertion is

$$\tau(k)=\left(1+a\lambda_0\log{\Lambda\over k}\right)^{-1/3}.$$

The physical mass is the inverse correlation length,

$$m=\xi^{-1}.$$

Thus $k\sim m$ is the scale where the critical RG flow stops.

Three mass parameters, three jobs

A reader will avoid many confusions by keeping the following three quantities separate.

1. $r_0$ is the bare coordinate in the microscopic action. It is the number one changes in the cutoff theory or lattice model.
2. $r_{0,c}$ is the critical coordinate. It is not universal; it depends on the cutoff scheme and on all other microscopic couplings.
3. $m=\xi^{-1}$ is the physical infrared scale. It is defined from the large-distance decay of correlation functions, or equivalently from the small-momentum inverse two-point function.

In the symmetric phase a convenient operational definition uses the one-particle-irreducible inverse propagator,

$$\Gamma^{(2)}(p)=Z_m\left(p^2+m^2+O(p^4/m^2)\right) \qquad (p\ll m),$$

so that

$$m^2={\Gamma^{(2)}(0)\over Z_m}.$$

Criticality means $\Gamma^{(2)}(0)=0$ after the local counterterms have been included. This is the sense in which tuning $r_0$ is not cosmetic: it moves the theory onto the critical surface.

At the Gaussian fixed point in $d$ Euclidean dimensions, a scalar field has engineering dimension

$$[\phi]={d-2\over2}.$$

Therefore

$$[\phi^2]=d-2, \qquad [\phi^4]=2(d-2), \qquad [\phi^6]=3(d-2).$$

The action is dimensionless, so the coupling multiplying a local operator $\mathcal O$ has engineering dimension $d-\Delta_{\mathcal O}$. In $d=4$,

$$[r_0]=2, \qquad [\lambda_0]=0, \qquad [c_6/\Lambda^2]=-2.$$

Thus the mass term is relevant, the quartic coupling is marginal by power counting, and a $\phi^6$ term is irrelevant. Written in terms of dimensionless couplings at scale $k$,

$$\widehat r(k)={r(k)\over k^2}, \qquad \widehat c_6(k)\sim c_6{k^2\over\Lambda^2},$$

coarse graining toward the infrared gives, at tree level,

$${d\widehat r\over dL}=2\widehat r, \qquad {d\widehat c_6\over dL}=-2\widehat c_6, \qquad L=\log{\Lambda\over k}.$$

The relevant variable grows as the theory is viewed at longer distances. The irrelevant variable shrinks. The quartic coupling is special in four dimensions: it is marginal at tree level, and loop effects make it run only logarithmically.

The relevant mass direction is why a critical point is not generic. In the space of bare local actions, the condition $m_{\rm phys}=0$ defines a surface of codimension one. Moving along this surface changes marginal and irrelevant couplings, but keeps the theory critical. Moving away from it produces a finite correlation length.

The critical surface is the set of bare actions whose physical mass vanishes. The deviation $t_0=r_0-r_{0,c}$ is a relevant coordinate: it grows under coarse graining and sends the theory into a massive phase with finite correlation length.

The critical value is not generally $r_0=0$. It is a function of the cutoff and of the other bare couplings,

$$r_{0,c}=r_{0,c}(\Lambda,\lambda_0,c_{6,0},\ldots).$$

The scaling variable is the deviation from this surface,

$$t_0=r_0-r_{0,c}.$$

In a magnetic system, $t_0$ is proportional to $T-T_c$ after a nonuniversal normalization. In a relativistic scalar theory, $t_0$ is the renormalized mass-squared parameter, again up to multiplicative logarithmic factors.

The physical mass stops the flow

At the critical point, there is no intrinsic infrared scale. If we compute a vertex at external momentum $q$, the logarithmic flow can run down to $k\sim q$. Away from the critical point, the long-distance two-point function is massive,

$$G(p)\simeq {Z\over p^2+m^2}$$

at small Euclidean momentum. The mass $m$ cuts off the critical singularities. For momenta $k\gg m$, the field still behaves approximately as a critical field; for $k\lesssim m$, the propagator no longer looks scale invariant.

Therefore the physically measured low-energy quartic coupling is naturally defined at the scale $k=m$:

$$\boxed{ \lambda_{\rm phys} =\lambda(m) ={\lambda_0\over 1+a\lambda_0\log(\Lambda/m)}. }$$

For intermediate scales $m\lesssim k\ll\Lambda$, the same running coupling can be expressed in terms of $\lambda_{\rm phys}$. Since

$${1\over\lambda(k)}={1\over\lambda_0}+a\log{\Lambda\over k},$$

and

$${1\over\lambda_{\rm phys}}={1\over\lambda_0}+a\log{\Lambda\over m},$$

we subtract the two equations and obtain

$${1\over\lambda(k)}={1\over\lambda_{\rm phys}}-a\log{k\over m}.$$

Equivalently,

$$\boxed{ \lambda(k)={\lambda_{\rm phys}\over 1-a\lambda_{\rm phys}\log(k/m)}. }$$

This formula is just the previous running coupling with a different boundary condition. Instead of specifying the bare coupling at the cutoff, we specify the physical coupling at the mass scale.

The critical running of the quartic coupling stops at the physical mass scale $m=\xi^{-1}$. The value reached there is the physical low-energy coupling $\lambda_{\rm phys}=\lambda(m)$.

There is an important consequence. If $\lambda_0>0$ is fixed and the cutoff is taken far above the physical mass, then

$$\lambda_{\rm phys} ={\lambda_0\over 1+a\lambda_0\log(\Lambda/m)} \longrightarrow 0 \qquad (\Lambda/m\to\infty).$$

At leading-log accuracy,

$$\lambda_{\rm phys}\sim {1\over a\log(\Lambda/m)}.$$

Thus the four-dimensional one-component $\phi^4$ theory approaches a free theory in this continuum limit. Conversely, if one insists on holding a positive finite $\lambda_{\rm phys}$ fixed, then

$${1\over\lambda_0}={1\over\lambda_{\rm phys}}-a\log{\Lambda\over m}.$$

For sufficiently large $\Lambda/m$, the right-hand side becomes negative. Perturbatively this is the Landau-pole obstruction written backward: the theory is perfectly useful as an effective field theory with a finite cutoff, but the interacting continuum limit is not obtained by taking $\Lambda/m\to\infty$ at fixed positive physical coupling.

Additive mass renormalization

The mass term is relevant not only in the infrared. It is also the local operator most easily generated by ultraviolet loops. This is why the critical value $r_{0,c}$ depends on the cutoff.

In the symmetric massive theory, take $m_0^2>0$ for the following explicit cutoff integral. In the theory

$$S_E\supset\int d^4x\left[{1\over2}m_0^2\phi^2+{\lambda_0\over4!}\phi^4\right],$$

the one-loop tadpole contributes a local self-energy

$$\delta m_1^2={\lambda_0\over2}\int_{|\ell|<\Lambda}{d^4\ell\over(2\pi)^4}{1\over \ell^2+m_0^2}.$$

With a spherical Euclidean cutoff,

$$\int_{|\ell|<\Lambda}{d^4\ell\over(2\pi)^4}{1\over \ell^2+m_0^2} ={1\over16\pi^2}\left[ \Lambda^2-m_0^2\log{\Lambda^2+m_0^2\over m_0^2} \right].$$

Thus, for $\Lambda\gg m_0$,

$$\delta m_1^2 ={\lambda_0\over32\pi^2}\Lambda^2 -{\lambda_0 m_0^2\over32\pi^2}\log{\Lambda^2\over m_0^2} +\cdots.$$

The coefficient of the quadratic term depends on the regulator, but the existence of an additive local mass shift does not. Higher loops generate further local terms of the schematic form

$$\delta m^2 =A_1\lambda_0\Lambda^2 +A_2\lambda_0^2\Lambda^2\log{\Lambda\over\mu} +\cdots,$$

where the constants $A_i$ depend on the cutoff convention and on how the local part is separated from the nonlocal part.

The physical mass is obtained after adding the bare quadratic term and local self-energy corrections. Criticality requires tuning this sum so that the pole or inverse correlation length satisfies $m_{\rm phys}=0$.

The physical mass is therefore not the bare parameter $m_0$. It is the location of the pole of the propagator, or in Euclidean statistical language the inverse correlation length. Near the critical point one may write schematically

$$m_{\rm phys}^2 =m_0^2+\text{local self-energy at zero momentum}.$$

The critical bare value is the one that makes this vanish:

$$m_0^2=m_{0,c}^2(\Lambda,\lambda_0,\ldots).$$

After subtracting the critical value, the remaining deviation

$$t_0=m_0^2-m_{0,c}^2$$

is the relevant scaling field. In a statistical system this tuning is ordinary experimental control: changing $T$ moves the system through $T_c$. In a fundamental scalar theory the same statement becomes the naturalness problem: a small physical scalar mass requires a cancellation between a bare parameter and cutoff-sensitive local quantum corrections.

Correlation length and logarithmic mass scaling

The previous page only needed the rough relation

$$m^2\propto |T-T_c|$$

inside a logarithm. That is enough for the specific-heat logarithm because constants and powers inside the logarithm change only subleading terms. But the mass itself also receives a logarithmic correction at the upper critical dimension.

Let

$$t_0=r_0-r_{0,c}$$

be the deviation from the critical surface. A mass perturbation inserts the thermal operator

$$\mathcal E={1\over2}\phi^2.$$

At momenta $p$ still above the physical mass, the leading-log dressed insertion carries the factor

$$\tau(p)=\left(1+a\lambda_0\log{\Lambda\over p}\right)^{-1/3}.$$

Thus the inverse propagator in the critical regime has the schematic form

$$G^{-1}(p;t_0)\simeq p^2+t_0\,\tau(p)+\cdots, \qquad m\ll p\ll\Lambda.$$

The flow stops when the two terms become comparable at $p\sim m$:

$$m^2\simeq t_0\,\tau(m).$$

Therefore

$$\boxed{ m^2\simeq t_0\left(1+a\lambda_0\log{\Lambda\over m}\right)^{-1/3}. }$$

Solving this relation asymptotically gives, up to nonuniversal constants inside the logarithm,

$$\boxed{ m^2\sim |t_0|\left[\log{1\over |t_0|}\right]^{-1/3}, \qquad \xi={1\over m}\sim |t_0|^{-1/2} \left[\log{1\over |t_0|}\right]^{1/6}. }$$

The mean-field exponent $\nu=1/2$ survives in four dimensions, but it is dressed by a universal power of a logarithm. The exponent $1/6$ is for the one-component scalar theory with the normalization used here. For an $O(N)$ model the power changes, but the logic does not: the relevant thermal direction fixes the leading power law, while the marginally irrelevant quartic coupling supplies a slow logarithmic dressing.

This is the same physics as the logarithmic specific heat: the Gaussian fixed point controls the power law, while the marginally irrelevant quartic coupling controls the logarithmic correction.

A finite external momentum, a finite physical mass, and a finite box size all act as infrared cutoffs. In a finite system of linear size $R$, the smallest momentum is of order $1/R$, so at leading-log accuracy the running stops at

$$k_{\rm IR}\sim \max(q,m,R^{-1}).$$

One then replaces

$$\log{\Lambda\over q} \quad\text{by}\quad \log{\Lambda\over k_{\rm IR}}.$$

Irrelevant operators and shifted critical data

The effective action contains every local operator allowed by the symmetries. The reason this does not destroy predictivity is the scaling hierarchy. In four dimensions,

$$\int d^4x\,{c_6\over\Lambda^2}\phi^6$$

has a dimensionless coefficient at scale $k$ of order

$$c_6(k)\sim c_6{k^2\over\Lambda^2}.$$

It dies in the infrared. Such an operator can still affect local quantities. For example, contracting fields inside $\phi^6$ can shift the coefficient of $\phi^4$, the coefficient of $\phi^2$, and therefore the numerical value of $T_c$. But after the critical point is retuned, it does not change the leading universal logarithms governed by the marginal quartic coupling and the relevant thermal scaling field.

Near the four-dimensional Gaussian fixed point, $t\phi^2$ is relevant, $\lambda\phi^4$ is marginal with logarithmic running, and $c_6\phi^6/\Lambda^2$ is irrelevant. Irrelevant operators can shift the critical surface but not the leading universal logarithms.

This also explains why dimensions above four behave differently. Consider the massless bubble integral in $d$ dimensions,

$$I_d(k)=\int^\Lambda {d^d\ell\over(2\pi)^d}{1\over \ell^2(\ell+k)^2}.$$

Dimensional analysis gives a nonlocal dependence

$$I_d(k)_{\rm nonlocal}\sim k^{d-4},$$

while local cutoff-dependent pieces are analytic in $k^2$ and are absorbed into local counterterms. In $d=4$, $k^{d-4}$ becomes a logarithm. In $d=5$, the dominant term is cutoff dependent and local, schematically

$$I_5(k)=A\Lambda+B|k|+\text{local analytic terms},$$

with no large logarithm. The quartic coupling has engineering dimension $4-d=-1$ and is irrelevant. The leading bulk critical powers are mean-field, and the four-dimensional logarithmic corrections are absent. The quartic coupling is nevertheless dangerously irrelevant above four dimensions: it is needed to stabilize the ordered phase and can enter amplitudes, finite-size scaling, and hyperscaling relations through inverse powers. “Irrelevant” describes its linearized flow near the Gaussian fixed point, not permission to delete it from every observable.

What has been separated

It is useful to separate three kinds of statements that are often mixed together.

First, the critical surface is determined by local UV-sensitive data. The value $r_{0,c}$ depends on the regulator, the cutoff, and irrelevant couplings. This dependence is not universal.

Second, the scaling variable is the deviation from the critical surface,

$$t_0=r_0-r_{0,c}.$$

This is the relevant parameter that determines the inverse correlation length.

Third, the universal long-distance behavior is determined by the RG flow near the fixed point. In four-dimensional $\phi^4$ theory, the quartic coupling is marginally irrelevant, so universal critical behavior contains logarithms rather than new non-mean-field powers.

This separation is the operational content of renormalization. UV fluctuations move the coordinates of the critical surface, while the RG flow near that surface controls universal long-distance singularities.

The next page turns this local viewpoint into an algebra of operators. Instead of asking only how couplings run, we ask what happens when two local fields approach one another. The answer is the operator product expansion.

Summary

The coefficient of $\phi^2$ is relevant. In four dimensions,

$$[r_0]=2,$$

so the dimensionless mass variable grows under coarse graining as the theory flows toward the infrared.

The condition for criticality is not $r_0=0$, but

$$r_0=r_{0,c}(\Lambda,\lambda_0,c_{6,0},\ldots),$$

or equivalently

$$m_{\rm phys}=0.$$

The deviation

$$t_0=r_0-r_{0,c}$$

is the relevant scaling variable.

A nonzero physical mass stops critical running at

$$k\sim m=\xi^{-1}.$$

The physical quartic coupling measured at that scale is

$$\lambda_{\rm phys}={\lambda_0\over 1+{3\lambda_0\over16\pi^2}\log(\Lambda/m)}.$$

Equivalently,

$$\lambda(k)={\lambda_{\rm phys}\over 1-{3\lambda_{\rm phys}\over16\pi^2}\log(k/m)}.$$

Local self-energy corrections shift the mass additively. With a hard cutoff,

$$\delta m_1^2={\lambda_0\over32\pi^2}\Lambda^2+\text{logarithmic and finite terms}.$$

This shift determines the nonuniversal location of the critical surface.

The leading-log relation between the physical mass and the thermal scaling variable is

$$m^2\simeq t_0 \left(1+{3\lambda_0\over16\pi^2}\log{\Lambda\over m}\right)^{-1/3},$$

so

$$\xi\sim |t_0|^{-1/2} \left[\log{1\over |t_0|}\right]^{1/6}.$$

Irrelevant operators such as $\phi^6/\Lambda^2$ can shift $T_c$ and other local data, but they do not change the leading universal logarithms controlled by the marginally irrelevant quartic coupling.

Common pitfalls

Do not identify the bare mass $m_0$ with the physical mass $m_{\rm phys}$. The latter is defined by the pole of the propagator or by the inverse correlation length.

Do not locate the critical point by setting the bare quadratic coefficient to zero. The correct condition is $r_0=r_{0,c}$, and $r_{0,c}$ generally contains cutoff-dependent loop corrections.

Do not run critical logarithms below the mass scale. Once $k\lesssim m$, the propagator is massive and the critical RG flow is cut off.

Do not treat the coefficient of a quadratic divergence as universal. A hard cutoff, a lattice cutoff, Pauli–Villars fields, and dimensional regularization organize local mass terms differently. The need to tune a relevant scalar mass is physical; the numerical coefficient of a cutoff-dependent local term is not universal.

Do not confuse irrelevant with nonexistent. Irrelevant operators affect the location of the critical surface and analytic background terms. They are irrelevant to leading infrared singularities, not irrelevant to every number in the microscopic theory.

Do not conclude from logarithmic triviality that the cutoff theory is useless. Four-dimensional $\phi^4$ theory is an excellent effective theory over a finite range of scales. The obstruction concerns the interacting continuum limit at fixed positive physical coupling.

Exercises

Exercise 1 — Gaussian operator classification

At the Gaussian fixed point in $d$ Euclidean dimensions, show that

$$[\phi]={d-2\over2},$$

and find the engineering dimensions of the couplings multiplying $\phi^2$, $\phi^4$, and $\phi^6$. Classify these operators in $d=4$.

Solution

The kinetic term is

$$\int d^dx\,{1\over2}(\partial\phi)^2.$$

Since the action is dimensionless, and $[d^dx]=-d$, we require

$$2+2[\phi]-d=0.$$

Therefore

$$[\phi]={d-2\over2}.$$

For an operator $\phi^n$,

$$[\phi^n]=n{d-2\over2}.$$

The coupling $g_n$ in

$$\int d^dx\,g_n\phi^n$$

has dimension

$$[g_n]=d-n{d-2\over2}.$$

Thus

$$[g_2]=2, \qquad [g_4]=4-d, \qquad [g_6]=6-2d.$$

In $d=4$,

$$[g_2]=2, \qquad [g_4]=0, \qquad [g_6]=-2.$$

Therefore $\phi^2$ is relevant, $\phi^4$ is marginal by engineering dimension, and $\phi^6$ is irrelevant.

Exercise 2 — The hard-cutoff tadpole

Evaluate the leading large-$\Lambda$ behavior of

$$I(m_0,\Lambda)=\int_{|\ell|<\Lambda}{d^4\ell\over(2\pi)^4}{1\over\ell^2+m_0^2}.$$

Use it to find the one-loop tadpole mass shift in $\lambda_0\phi^4/4!$ theory.

Solution

In four Euclidean dimensions,

$$d^4\ell=2\pi^2\ell^3d\ell.$$

Therefore

$$I(m_0,\Lambda) ={1\over(2\pi)^4}2\pi^2\int_0^\Lambda d\ell\,{\ell^3\over \ell^2+m_0^2}.$$

Set $u=\ell^2$, so $\ell^3d\ell={1\over2}u\,du$. Then

$$I(m_0,\Lambda) ={1\over16\pi^2}\int_0^{\Lambda^2}du\,{u\over u+m_0^2}.$$

Since

$${u\over u+m_0^2}=1-{m_0^2\over u+m_0^2},$$

we get

$$I(m_0,\Lambda) ={1\over16\pi^2}\left[ \Lambda^2-m_0^2\log{\Lambda^2+m_0^2\over m_0^2} \right].$$

For $\Lambda\gg m_0$,

$$I(m_0,\Lambda) ={\Lambda^2\over16\pi^2} -{m_0^2\over16\pi^2}\log{\Lambda^2\over m_0^2}+\cdots.$$

In $\lambda_0\phi^4/4!$ theory, the one-loop tadpole self-energy has the combinatorial factor $\lambda_0/2$, so

$$\delta m_1^2={\lambda_0\over2}I(m_0,\Lambda).$$

Thus

$$\delta m_1^2 ={\lambda_0\over32\pi^2}\Lambda^2 -{\lambda_0m_0^2\over32\pi^2}\log{\Lambda^2\over m_0^2} +\cdots.$$

The quadratic term is cutoff-scheme dependent and local. It is absorbed into the location of the critical surface.

Exercise 3 — Matching at the physical mass

Let

$$\lambda(k)={\lambda_0\over1+a\lambda_0\log(\Lambda/k)}$$

and define

$$\lambda_{\rm phys}=\lambda(m).$$

Show that

$$\lambda(k)={\lambda_{\rm phys}\over1-a\lambda_{\rm phys}\log(k/m)}.$$

What happens to $\lambda_{\rm phys}$ as $\Lambda/m\to\infty$ with fixed positive $\lambda_0$?

Solution

Invert the running coupling:

$${1\over\lambda(k)}={1\over\lambda_0}+a\log{\Lambda\over k}.$$

At $k=m$,

$${1\over\lambda_{\rm phys}}={1\over\lambda_0}+a\log{\Lambda\over m}.$$

Subtract the second equation from the first:

$${1\over\lambda(k)}-{1\over\lambda_{\rm phys}} =a\left(\log{\Lambda\over k}-\log{\Lambda\over m}\right) =-a\log{k\over m}.$$

Therefore

$${1\over\lambda(k)}={1\over\lambda_{\rm phys}}-a\log{k\over m},$$

or

$$\lambda(k)={\lambda_{\rm phys}\over1-a\lambda_{\rm phys}\log(k/m)}.$$

For fixed positive $\lambda_0$,

$$\lambda_{\rm phys} ={\lambda_0\over1+a\lambda_0\log(\Lambda/m)}.$$

As $\Lambda/m\to\infty$, the denominator grows without bound, so

$$\lambda_{\rm phys}\to0.$$

This is the leading-log form of triviality for the continuum limit of positive four-dimensional $\phi^4$ theory.

Exercise 4 — The correlation-length logarithm

Assume the dressed thermal insertion is

$$\tau(k)=\left(1+a\lambda_0\log{\Lambda\over k}\right)^{-1/3},$$

and that the physical mass is determined by

$$m^2\simeq t_0\tau(m).$$

Derive the leading logarithmic behavior of the correlation length $\xi=1/m$ as $t_0\to0$.

Solution

The defining relation is

$$m^2\simeq t_0\left(1+a\lambda_0\log{\Lambda\over m}\right)^{-1/3}.$$

As $t_0\to0$, the mass becomes small, so the logarithm is large. To leading logarithmic accuracy, the mean-field relation gives $m\sim |t_0|^{1/2}$ inside the logarithm. Therefore

$$\log{\Lambda\over m} = {1\over2}\log{1\over |t_0|}+\text{subleading constants and logarithms}.$$

Thus

$$m^2\sim |t_0|\left[\log{1\over |t_0|}\right]^{-1/3}.$$

Taking the square root,

$$m\sim |t_0|^{1/2}\left[\log{1\over |t_0|}\right]^{-1/6}.$$

Since $\xi=1/m$,

$$\boxed{ \xi\sim |t_0|^{-1/2} \left[\log{1\over |t_0|}\right]^{1/6}. }$$

The power $1/2$ is the mean-field exponent, while $1/6$ is the logarithmic correction exponent for the one-component theory.

Exercise 5 — How an irrelevant operator shifts critical data

In four dimensions, consider the irrelevant perturbation

$$\Delta S=\int d^4x\,{c_6\over\Lambda^2}\phi^6.$$

Show by power counting that its dimensionless strength at scale $k$ is of order $c_6(k/\Lambda)^2$. Explain why it can shift $T_c$ but not the leading critical logarithms.

Solution

In four dimensions,

$$[\phi]=1, \qquad [\phi^6]=6.$$

The coupling multiplying $\phi^6$ must have dimension

$$4-6=-2.$$

Writing it as $c_6/\Lambda^2$ makes $c_6$ dimensionless at the cutoff scale. At a lower scale $k$, the natural dimensionless strength is obtained by multiplying by $k^2$:

$$g_6(k)\sim {c_6\over\Lambda^2}k^2=c_6\left({k\over\Lambda}\right)^2.$$

Thus the perturbation becomes small in the infrared.

However, loops involving $\phi^6$ can contract fields at short distances and generate local lower-dimension operators, including $\phi^4$ and $\phi^2$. These contributions shift nonuniversal quantities such as the critical value $r_{0,c}$ and hence $T_c$. After the critical surface is retuned, the leading long-distance logarithms are controlled by the marginal quartic coupling and the thermal scaling field, not by $\phi^6$.

Further reading

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, chapters 22–23.
• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, sections 28–29.
• Weinberg, Steven. The Quantum Theory of Fields, Volume II: Modern Applications. Cambridge University Press, 1996, sections 18.4–18.5.
• Wilson, Kenneth G., and John Kogut. “The Renormalization Group and the $\epsilon$ Expansion.” Physics Reports 12 (1974): 75–199.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 4th ed., Oxford University Press, 2002, chapters 8–19.