The perturbative all-loop derivation of the NSVZ
β
-function for
N
=
1
supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities ...produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the
β
-function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the
β
-function and these anomalous dimensions for the renormalization group functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD + MSL prescription.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
A
bstract
For a general
N
= 1 supersymmetric gauge theory regularized by higher covariant derivatives we prove in all orders that the
β
-function defined in terms of the bare couplings is given by ...integrals of double total derivatives with respect to loop momenta. With the help of the technique used for this proof it is possible to construct a method for obtaining these loop integrals, which essentially simplifies the calculations. As an illustration of this method, we find the expression for the three-loop contribution to the
β
-function containing the Yukawa couplings and compare it with the result of the standard calculations made earlier. Also we briefly discuss, how the structure of the loop integrals for the
β
-function considered in this paper can be used for the all-loop perturbative derivation of the NSVZ relation in the non-Abelian case.
A
bstract
The contributions of the matter superfields and of the Faddeev-Popov ghosts to the
β
-function of
N
= 1 supersymmetric gauge theories defined in terms of the bare couplings are calculated ...in all orders in the case of using the higher covariant derivative regularization. For this purpose we use the recently proved statement that the
β
-function in these theories is given by integrals of double total derivatives with respect to the loop momenta. These integrals do not vanish due to singularities of the integrands. This implies that the
β
-function beyond the one-loop approximation is given by the sum of the singular contributions, which is calculated in all orders for singularities produced by the matter superfields and by the Faddeev-Popov ghosts. The result is expressed in terms of the anomalous dimensions of these superfields. It coincides with the corresponding part of the new form of the NSVZ equation, which can be reduced to the original one with the help of the non-renormalization theorem for the triple gauge-ghost vertices.
We consider a one-loop finite
N
=
1
supersymmetric theory in such a renormalization scheme that the first
L
contributions to the gauge
β
-function and the first
(
L
-
1
)
contributions to the ...anomalous dimension of the matter superfields and to the Yukawa
β
-function vanish. It is demonstrated that in this case the NSVZ equation and the exact equation for the Yukawa
β
-function in the first nontrivial order are valid for an arbitrary renormalization prescription respecting the above assumption. This implies that under this assumption the
(
L
+
1
)
-loop contribution to the gauge
β
-function and the
L
-loop contribution to the Yukawa
β
-function are always expressed in terms of the
L
-loop contribution to the anomalous dimension of the matter superfields. This statement generalizes the result of Grisaru, Milewski, and Zanon that for a theory finite in
L
loops the
(
L
+
1
)
-loop contribution to the
β
-function also vanishes. In particular, it gives a simple explanation why their result is valid although the NSVZ equation does not hold in an arbitrary subtraction scheme.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
A
bstract
Three-loop
β
-functions of the Minimal Supersymmetric Standard Model regularized by higher covariant derivatives are obtained for an arbitrary supersymmetric subtraction scheme. For this ...purpose we first calculate two-loop anomalous dimensions for all MSSM chiral matter superfields defined in terms of the bare couplings. Then we use the NSVZ equations for the renormalization group functions defined in terms of the bare couplings, which are valid in all orders in the case of using the higher covariant derivative regularization. This gives the three-loop
β
-functions defined in terms of the bare couplings. After that, we construct the three-loop
β
-functions and the two-loop anomalous dimensions standardly defined in terms of the renormalized couplings for an arbitrary subtraction scheme. As a nontrivial correctness test, we verify that for a certain renormalization prescription the general results reproduce the ones obtained earlier in the
DR
¯
scheme. Also this can be considered as an indepedent confirmation of the
DR
¯
results.
Using the Slavnov–Taylor identities we prove that the three-point ghost vertices with a single line of the quantum gauge superfield are not renormalized in all loops in N=1 supersymmetric gauge ...theories. This statement is verified by the explicit one-loop calculation made by the help of the BRST invariant version of the higher covariant derivative regularization. Using the restrictions to the renormalization constants which are imposed by the non-renormalization of the considered vertices we express the exact NSVZ β-function in terms of the anomalous dimensions of the Faddeev–Popov ghosts and of the quantum gauge superfield. In the expression for the NSVZ β-function obtained in this way the contributions of the Faddeev–Popov ghosts and of the matter superfields have the same structure.
A
bstract
For
N
= 1 SQED with
N
f
flavors regularized by higher derivatives in the general
ξ
-gauge we calculate the three-loop anomalous dimension of the matter superfields defined in terms of the ...bare coupling constant and demonstrate its gauge independence. After this the four-loop
β
-function defined in terms of the bare coupling constant is obtained with the help of the NSVZ equation, which is valid for these renormalization group functions in all loops. Next, we calculate the three-loop anomalous dimension and the four-loop
β
-function defined in terms of the renormalized coupling constant for an arbitrary subtraction scheme supplementing the higher derivative regularization. Then we construct a renormalization prescription for which the results coincide with the ones in the
DR
¯
-scheme and describe all NSVZ schemes in the considered approximation. Also we demonstrate the existence of a subtraction scheme in which the anomalous dimension does not depend on
N
f
, while the
β
-function contains only terms of the first order in
N
f
. This scheme is obtained with the help of a finite renormalization compatible with a structure of quantum corrections and is NSVZ. The existence of such an NSVZ scheme is also proved in all loops.
A
bstract
The two-loop anomalous dimension of the chiral matter superfields is calculated for a general
N
= 1 supersymmetric gauge theory regularized by higher covariant derivatives. We obtain both ...the anomalous dimension defined in terms of the bare couplings, and the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription. For the one-loop finite theories we find a simple relation between the higher derivative regulators under which the anomalous dimension defined in terms of the bare couplings vanishes in the considered approximation. In this case the one-loop finite theory is also two-loop finite in the HD+MSL scheme. Using the assumption that with the higher covariant derivative regularization the NSVZ equation is satisfied for RGFs defined in terms of the bare couplings, we construct the expression for the three-loop
β
-function. Again, the result is written both for the
β
-function defined in terms of the bare couplings and for the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription.
A
bstract
We consider a version of dimensional regularization (reduction) in which the dimensionful regularization parameter Λ is in general different from the renormalization scale
μ
. Then in the ...scheme analogous to the minimal subtraction the renormalization constants contain
ε
-poles, powers of ln Λ
/μ
, and mixed terms of the structure
ε
−q
ln
p
Λ
/μ
. For the MS-like schemes we present explicit expressions for the coefficients at all these structures which relate them to the coefficients in the renormalization group functions, namely in the
β
-function and in the anomalous dimension. In particular, for the pure
ε
-poles we present explicit solutions of the ’t Hooft pole equations. Also we construct simple all-loop expressions for the renormalization constants (also written in terms of the renormalization group functions) which produce all
ε
-poles and logarithms and establish a number of relations between various coefficients at
ε
-poles and logarithms. The results are illustrated by some examples.
For a general renormalizable N = 1 supersymmetric gauge theory with a simple gauge group we verify the ultraviolet (UV) finiteness of the two-loop matter contribution to the triple gauge-ghost ...vertices. These vertices have one leg of the quantum gauge superfield and two legs corresponding to the Faddeev–Popov ghost and antighost. By an explicit calculation made with the help of the higher covariant derivative regularization we demonstrate that the sum of the corresponding two-loop supergraphs containing a matter loop is not UV divergent in the case of using a general ξ -gauge. In the considered approximation this result confirms the recently proved theorem that the triple gauge-ghost vertices are UV finite in all orders, which is an important ingredient of the all-loop perturbative derivation of the Novikov-Shifman-Vainshtein-Zakharov relation.
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