- Email: [email protected]
Anomalous dimensions of scalar operators in CFT
Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 197–198 www.elsevierphysics.com
Anomalous dimensions of scalar operators in CFT Alessandro Vichi a a
Institut de Th´eorie des Ph´enom`enes Physiques, EPFL, CH–1015 Lausanne, Switzerland
We present a recently developed method to constrain the anomalous dimension of scalar operators in a general Conformal Field Theory (CFT). Using a consistency condition derived from four-point correlation functions it is possible to bind the anomalous dimension of a composite operator φ2 . The result has also consequences on model building.
1. Introduction Recent proposals in model building exploit large anomalous dimensions of operators in order to make irrelevant, or at least marginal, terms that would otherwise be dangerous. This is the case, for example, of Conformal Technicolor proposed in ([1]). In that model the stability of the weak scale and the absence of unwanted Flavor violating effects translate into two conditions on the scaling dimension of the Higgs field φ and of the composite operator φ2 : [φ] ∼ 1 , φ2 ∼ 4 (a more precise discussion can be found in [2]). These cannot be achieved without dealing with strongly coupled theories, which however are not easy to handle. Moreover, because of unitarity constraints, a scalar field in four dimensions can have dimension equal to 1 only if it is free, which therefore implies [φ2 ] = 2. It would therefore be interesting to make a quantitative statement on how close a field theory can come to realize the above conditions. Another way to phrase the question is: what can we say on the anomalous dimension of φ when the anomalous dimension of φ2 is small?
2. The result In a 2 or 4 dimensional CFT, consider a scalar operator φ with anomalous dimension d and call φ2 the lowest dimension scalar operator appearing with non vanishing coefficient in the OPE φ×φ. Starting from first principles, such as unitarity, crossing symmetry and conformal invariance, 0920-5632/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2009.07.079
4.0
gd
fd
3.5 3.0 2.5 2.0 d 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35
Figure 1. Bound for a 4 dimensional CFT. Dashed line: analytic bound; blue line: best numerical refinement up to now.
it is possible to prove that there exists an upper bound to the anomalous dimension Δ of the operator φ2 : Δ < f (d). The best estimate of f (d) up to now has been obtained numerically, while a weaker constraint Δ < g(d), with g(d) > f (d), can be obtained analytically. All the results presented here are formally derived and discussed in details in [2]. 2.1. Four dimensions Figure (1) summarizes the result: given a real scalar field φ with dimension d, there exists no four dimensional CFT where the dimension Δ of φ2 is bigger than f (d). The figure shows the bound starting with a square root behavior and reaching a linear one after d − 1 > 0.1:
198
A. Vichi / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 197–198
√ f (d) = 2 + 1.79 d − 1 + 2.9(d − 1). The result implies a continuity in reaching the free theory regime, which translates into the impossibility of having arbitrary large anomalous dimensions for the composite operator if the elementary one has not. In four dimensions very few examples of calculable CFTs are known and they don’t lead to interesting comparisons with the bound since they lie abundantly below it. This is because the bound is significantly above the line Δ = 2d while no known model exhibits the same property. The only exception is represented by the Wilson-Fischer fixed point which does display a square root behavior although it slightly violates the bound; however this model involves an extension of the theory in 4 − ε dimensions that can give rise to subtleties. 2.2. Two dimensions The two dimensional result is very encouraging. The existence of exact solutions far from the weakly coupled regime, such as the the Ising Model and other Minimal Models, lets comparisons to be performed. Despite the shown analytical bound (fig. 2) doesn’t converge to the origin, it is nearly saturated by several Minimal Models in the region d > 0.1. This suggests that there are values of d for which the method used to derive the bound is able to capture important features of a CFT. Recent results improve the bound in the region of small d and make it pass through the origin ([3]). On the contrary, the bound in four dimensions becomes stronger in the region of larger d while it keeps the same square root behavior near the origin. 3. The method In a quantum field theory the invariance under the conformal symmetry fixes the form of the 2 and 3-point correlation functions but leaves some arbitrariness in the 4-point one. Given an operator φ it is possible to show that the correlation function φ(x1 )φ(x2 )φ(x3 )φ(x4 ) encodes all the information concerning the dimension δ and the spin l of the operators appearing in the OPE φ × φ. Studing this correlation function one can
2.0 1.5 1.0
Ising
0.5 0.0
0.1
0.2
0.3
0.4
d
Figure 2. Bound for a 2 dimensional CFT. Black points: known anomalous dimensions of Minimal Models. Red point (0.125, 1): Ising models.
derive non trivial constraints on the structure of the theory. In particular, imposing some crossing symmetry one can obtain the following sum rule: Cl,δ F (d, l, δ, u, v) = 1 l ∈ N, even, δ ∈ R, (1) l,δ
where u, v are the harmonic ratios built from the xi . In the above equation F (d, l, δ, u, v) are known functions while Cl,δ are positive coefficients derived from the OPE φ × φ. The problem can be restated in the following way. Call Δ the minimal value of δ appearing in (1): then it is possible to find a function f (d) such that for Δ > f (d) the sum rule cannot be satisfied for any choice of the coefficient Cl,δ . The bound Δ < f (d) is a necessary but not sufficient condition for the existence of a CFT. In the space of the possible theories it delimits the region where (1) can be satisfied and has been obtained studying the behavior of the functions F (δ, l, d, u, v). Although improvements are possible and under study, there are values of d where the present bound seems already optimal. REFERENCES 1. Luty and Okui, JHEP 0609 (2006) 070 2. Rattazzi, Rychkov, Tonni and Vichi, JHEP 0812 (2008) 031 3. Rychkov, in preparation.
Anomalous dimensions of scalar operators in CFT Alessandro Vichi a a
Institut de Th´eorie des Ph´enom`enes Physiques, EPFL, CH–1015 Lausanne, Switzerland
We present a recently developed method to constrain the anomalous dimension of scalar operators in a general Conformal Field Theory (CFT). Using a consistency condition derived from four-point correlation functions it is possible to bind the anomalous dimension of a composite operator φ2 . The result has also consequences on model building.
1. Introduction Recent proposals in model building exploit large anomalous dimensions of operators in order to make irrelevant, or at least marginal, terms that would otherwise be dangerous. This is the case, for example, of Conformal Technicolor proposed in ([1]). In that model the stability of the weak scale and the absence of unwanted Flavor violating effects translate into two conditions on the scaling dimension of the Higgs field φ and of the composite operator φ2 : [φ] ∼ 1 , φ2 ∼ 4 (a more precise discussion can be found in [2]). These cannot be achieved without dealing with strongly coupled theories, which however are not easy to handle. Moreover, because of unitarity constraints, a scalar field in four dimensions can have dimension equal to 1 only if it is free, which therefore implies [φ2 ] = 2. It would therefore be interesting to make a quantitative statement on how close a field theory can come to realize the above conditions. Another way to phrase the question is: what can we say on the anomalous dimension of φ when the anomalous dimension of φ2 is small?
2. The result In a 2 or 4 dimensional CFT, consider a scalar operator φ with anomalous dimension d and call φ2 the lowest dimension scalar operator appearing with non vanishing coefficient in the OPE φ×φ. Starting from first principles, such as unitarity, crossing symmetry and conformal invariance, 0920-5632/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2009.07.079
4.0
gd
fd
3.5 3.0 2.5 2.0 d 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35
Figure 1. Bound for a 4 dimensional CFT. Dashed line: analytic bound; blue line: best numerical refinement up to now.
it is possible to prove that there exists an upper bound to the anomalous dimension Δ of the operator φ2 : Δ < f (d). The best estimate of f (d) up to now has been obtained numerically, while a weaker constraint Δ < g(d), with g(d) > f (d), can be obtained analytically. All the results presented here are formally derived and discussed in details in [2]. 2.1. Four dimensions Figure (1) summarizes the result: given a real scalar field φ with dimension d, there exists no four dimensional CFT where the dimension Δ of φ2 is bigger than f (d). The figure shows the bound starting with a square root behavior and reaching a linear one after d − 1 > 0.1:
198
A. Vichi / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 197–198
√ f (d) = 2 + 1.79 d − 1 + 2.9(d − 1). The result implies a continuity in reaching the free theory regime, which translates into the impossibility of having arbitrary large anomalous dimensions for the composite operator if the elementary one has not. In four dimensions very few examples of calculable CFTs are known and they don’t lead to interesting comparisons with the bound since they lie abundantly below it. This is because the bound is significantly above the line Δ = 2d while no known model exhibits the same property. The only exception is represented by the Wilson-Fischer fixed point which does display a square root behavior although it slightly violates the bound; however this model involves an extension of the theory in 4 − ε dimensions that can give rise to subtleties. 2.2. Two dimensions The two dimensional result is very encouraging. The existence of exact solutions far from the weakly coupled regime, such as the the Ising Model and other Minimal Models, lets comparisons to be performed. Despite the shown analytical bound (fig. 2) doesn’t converge to the origin, it is nearly saturated by several Minimal Models in the region d > 0.1. This suggests that there are values of d for which the method used to derive the bound is able to capture important features of a CFT. Recent results improve the bound in the region of small d and make it pass through the origin ([3]). On the contrary, the bound in four dimensions becomes stronger in the region of larger d while it keeps the same square root behavior near the origin. 3. The method In a quantum field theory the invariance under the conformal symmetry fixes the form of the 2 and 3-point correlation functions but leaves some arbitrariness in the 4-point one. Given an operator φ it is possible to show that the correlation function φ(x1 )φ(x2 )φ(x3 )φ(x4 ) encodes all the information concerning the dimension δ and the spin l of the operators appearing in the OPE φ × φ. Studing this correlation function one can
2.0 1.5 1.0
Ising
0.5 0.0
0.1
0.2
0.3
0.4
d
Figure 2. Bound for a 2 dimensional CFT. Black points: known anomalous dimensions of Minimal Models. Red point (0.125, 1): Ising models.
derive non trivial constraints on the structure of the theory. In particular, imposing some crossing symmetry one can obtain the following sum rule: Cl,δ F (d, l, δ, u, v) = 1 l ∈ N, even, δ ∈ R, (1) l,δ
where u, v are the harmonic ratios built from the xi . In the above equation F (d, l, δ, u, v) are known functions while Cl,δ are positive coefficients derived from the OPE φ × φ. The problem can be restated in the following way. Call Δ the minimal value of δ appearing in (1): then it is possible to find a function f (d) such that for Δ > f (d) the sum rule cannot be satisfied for any choice of the coefficient Cl,δ . The bound Δ < f (d) is a necessary but not sufficient condition for the existence of a CFT. In the space of the possible theories it delimits the region where (1) can be satisfied and has been obtained studying the behavior of the functions F (δ, l, d, u, v). Although improvements are possible and under study, there are values of d where the present bound seems already optimal. REFERENCES 1. Luty and Okui, JHEP 0609 (2006) 070 2. Rattazzi, Rychkov, Tonni and Vichi, JHEP 0812 (2008) 031 3. Rychkov, in preparation.