Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://nuclphys.sinp.msu.ru/conf/epp10/Shrock.pdf Äàòà èçìåíåíèÿ: Sat Sep 7 16:37:30 2013 Äàòà èíäåêñèðîâàíèÿ: Fri Feb 28 02:22:16 2014 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ultraviolet

Higher-Loop Calculations of the UV to IR Evolution of Gauge Theories and Remarks on Neutrino Properties

Robert Shro ck YITP, Stony Bro ok University

16'th Lomonosov Conference on Elementary Particle Physics, Moscow State Univ. 22-28 Aug. 2013


Outline
· Neutrino masses and lo op calculations of electromagnetic properties · Renormalization-group flow in an asymptotically free gauge theory from UV to IR; types of IR behavior; role of an exact or approximate IR fixed point · Higher-lo op calculations of UV to IR evolution, including IR zero of and anomalous dimension m of fermion bilinear · Some comparisons with lattice measurements of
m

· Higher-lo op calculation of structural properties of · Results in the limit Nc , Nf with Nf /Nc fixed · Study of scheme-dependence · Conclusions


Neutrino masses and loop calculations of electromagnetic properties
Neutrino masses and lepton mixing are confirmed evidence of physics beyond the Standard Mo del (SM), pioneering work by Pontecorvo; Gribov and Pontecorvo; Maki, Nakagawa, Sakata on searches for these. Neutrino oscillations discovered in solar and atmospheric experiments (Davis Homestake exp., Kamiokande, IMB, SAGE, GALLEX, SuperKamiokande, SNO) further studies via accelerator exps. (K2K, MINOS, MiniBo oNE, OPERA, T2K, NOvA...) and reactor exps. (KamLAND, Double Cho oz, Daya Bay, RENO...); great progress in gaining knowledge of masses and lepton mixing, with intensive current and future exp. programs; many talks at this conf. One extends the SM to include neutrino masses via addition of electroweak-singlet i,R fields, and hence Dirac and Majorana mass terms with interesting connection to possible UV completions of SM. Via seesaw mechanism, very small masses are plausibly related to very high mass scales of new physics beyond SM.


Questions still to be answered include mass hierarchy, leptonic CP violation, possibility of light, primarily electroweak-singlet (sterile) neutrinos (LSND,..) In addition to µ e , µ and Sanda, Lee, pro duces extrem oscillations, masses and lepton mixing lead to decays such as eee (in m -extended SM, Bilenky, Petcov, and Pontecorvo, Marciano ¯ Pakvasa, Shro ck, Sugawara (1977)) However, leptonic GIM mechanism ely small branching ratio.

Even with zero electric charge, neutrinos have induced diagonal and off-diagonal interactions with photons via one-lo op diagrams. These can mediate one-lo op induced decays (e.g., Shro ck (1974); 1977 papers...) Dirac neutrino has a magnetic dipole moment (Fujikawa and Shro ck, 1980; from general one-lo op formulas) 3eGF m m -19 µ µ = = (3.2 â 10 ) 22 1 eV 8

B

where µB = e/(2me). can also have CPV electric dipole moment. Majorana has transition dipole moments (diagonal ones vanish).


Detailed studies of effects on neutrino scattering - Studenikin, Voloshin,... Limits from reactor e exps (Savannah River, Krasnoyarsk, Rovno, MUNU, TEXONO, ¯ GEMMA..): µ < (3 â 10-11)µB ; also limits from LSND, SuperK, Borexino... Astrophysical limit from co oling of red giants: µ < (3 â 10-12)µB . Recent review of neutrino EM properties: Broggini, Giunti, Studenikin, arXiv:1207.3980. Searches for masses from nuclear beta decay exps (LANL, Tokyo, Zurich, Mainz, Moscow-Troitsk) continue to reduce limits on [ i |Uei|2m(i)2]1/2 to < 2 eV (future, KATRIN...). Stringent cosmological upper limit on i m(i). Also useful to search for emission of heavier neutrinos via mixing (Shro ck, Phys. Lett. B96, 159 (1980); Kobzarev, Martemyanov, Okun, Shchepkin, Yad. Fiz. 32, 1590 (1980) [Sov. J. Nucl. Phys. 32, 823 (1980)] via kink in Kurie plot, correlated upper limits on |Uei|2 from many searches; e.g., recent Troitsk limits in Belesev et al., JETP Letts. 97, 67 (2013) [arXiv:1211.7193], Belesev et al., arXiv:1307.5687. Neutrino masses, mixing, and electromagnetic properties of neutrinos continue to be important areas of theoretical and experimental study, esp ecially since they probe new physics beyond the SM.


The rest of this talk includes material from the following papers, and some new results: · Ryttov and Shro ck, "Higher-Lo op Corrections to the Infrared Evolution of a Gauge Theory with Fermions", Phys. Rev. D 83, 056011 (2011), arXiv:1011.4542 · Ryttov and Shro ck, "Scheme Transformations in the Vicinity of an Infrared Fixed Point", Phys. Rev. D 86, 065032 (2012), arXiv:1206.2366; "An Analysis of Scheme Transformations in the Vicinity of an Infrared Fixed Point", Phys. Rev. D 86, 085005 (2012), arXiv:1206.6895 · Shro ck, "Higher-Lo op Structural Properties of the Function in Asymptotically Free Vectorial Gauge Theories", Phys. Rev. D 87, 105005 (2013), arXiv:1301.3209 · Shro ck, "Higher-Lo op Calculations of the Ultraviolet to Infrared Evolution of a Vectorial Gauge Theory in the Limit Nc , Nf with Nf /Nc Fixed", Phys. Rev. D 87, 116007 (2013), arXiv:1302.5434 · Shro ck, `Study of Scheme Transformations to Remove Higher-Lo op Terms in the Function of a Gauge Theory", Phys. Rev. D 88, 036003 (2013), arXiv:1305.6524.


RG Flow from UV to IR; Types of IR Behavior and Role of IR Fixed Point
Consider an asymptotically free, vectorial gauge theory with gauge group G and N massless fermions in representation R of G.
f

Asymptotic freedom theory is weakly coupled, properties are perturbatively calculable for large Euclidean momentum scale µ in deep ultraviolet (UV). The quark-parton picture is applicable here and leads to scaling behavior in deep inelastic scattering, and quark-counting rules for d /dt and F (t) (Matveev, Muradyan, and Takhelidze; Bro dsky and Farrar). The question of how an asymptotically free gauge theory flows from large µ in the UV to small µ in the infrared (IR) depends on Nf and is of fundamental field-theoretic interest. In QCD with Nf = 2 or Nf = 3 light quarks, the function has no perturbative IR zero. We fo cus on a theory with larger Nf and hence different properties, where there may be an exact or approximate IR fixed point (zero of ).


Denote running gauge coupling at scale µ as g = g (µ), and let (µ) = g (µ)2/(4 ) and a(µ) = g (µ)2 /(16 2) = (µ)/(4 ). The dependence of (µ) on µ is described by the renormalization group (Stueckelberg and Peterman, Gell-Mann and Low, Bogoliubov, Shirkov, Callan, Symanzik, Wilson). The function is d = -2 dt
=1

b a = - 2

=1

¯ , b

where t = ln µ, = lo op order of the co eff. b, and ¯ = b/(4 ). b Co efficients b1 and b2 in are independent of regularization/renormalization scheme, while b for 3 are scheme-dependent. Asymptotic freedom means b1 > 0, so < 0 for small (µ), in neighborho o d of UV fixed point (UVFP) at = 0. As the scale µ decreases from large values, (µ) increases. Denote cr as minimum value for formation of bilinear fermion condensates and resultant spontaneous chiral symmetry breaking (SSB).


Two generic possibilities for and resultant UV to IR flow: · has no IR zero, so as µ decreases, (µ) increases, eventually beyond the perturbatively calculable region. This is the case for QCD. · has a IR zero, I R, so as µ decreases, I R . In this class of theories, there are two further generic possibilities: I R < cr or I R > cr . If I R < cr , the zero of at I R is an exact IR fixed point (IRFP) of the renorm. group (RG); as µ 0 and I R, (I R) = 0, and the theory becomes exactly scale-invariant with nontrivial anomalous dimensions. If has no IR zero, or an IR zero at I R > cr , then as µ decreases through a scale , (µ) exceeds cr and SSB o ccurs, so fermions gain dynamical masses . If SSB o ccurs, then in low-energy effective field theory applicable for µ < , one integrates these fermions out, and fn. becomes that of a pure gauge theory, with no IR zero. Hence, if has a zero at I R > cr , this is only an approx. IRFP of RG.


If I R is only slightly greater than cr , then, as (µ) approaches I R, since = d/dt 0, (µ) varies very slowly as a function of the scale µ, i.e., there is approximately scale-invariant (= dilatation-invariant) behavior. SSB at also breaks the approx. dilatation symmetry, might lead to a resultant approx. NGB, the dilaton. This is not massless, since (cr ) is nonzero. Denote the n-lo op fn. as At the n = 2 lo op level, I
R,2 n

and the IR zero of

n

as I

R,n

.

=-

4 b b
2

1

which is physical for b2 < 0. One-lo op co efficient b1 is (Gross and Wilczek, Politzer) b1 = (11CA - 4Nf Tf ) 3 where CA C2(G) is quadratic Casimir invariant, Tf T (R) is trace invariant. Fo cus here on G = SU(Nc). 1


Asymp. freedom requires Nf < N

f ,b1z

, where = 11C 4T
A

N

f ,b1z

f

e.g., for R = fundamental rep., Nf < (11/2)Nc. Two-lo op co eff. b2 is (with Cf C2(R)) (Caswell, Jones) b2 = 1 3
2 34CA - 4(5CA + 3Cf )Nf T f

For small Nf , b2 > 0; b2 decreases as fn. of Nf and vanishes with sign reversal at Nf = Nf ,b2z , where 2 34CA Nf ,b2z = 4Tf (5CA + 3Cf ) For arbitrary G and R, Nf ,b2z < N which has an IR zero, namely I: N
f ,b1z

, so there is always an interval in Nf for < Nf < N
f ,b1z

f ,b2z


· for SU(2), I : · for SU(3), I :

5.5 5 < N f < 1 1 8.05 < Nf < 16.5 2.62Nc < Nf < 5.5Nc.

· As Nc , I :

(expressions evaluated for Nf R, but it is understo o d that physical values of Nf are nonnegative integers.) As Nf decreases from the upper to lower end of interval I , I Nf = N
f ,cr R

increases. Denote

at I

R

= cr

Value of Nf ,cr is of fundamental importance, since it separates the (zero-temp.) chirally symmetric and broken IR phases. Intensive current lattice studies of SU(Nc) gauge theories with Nf copies of fermions in various representations R; progress toward determining Nf ,cr for various Nc and R.


Higher-Loop Corrections to UV IR Evolution of Gauge Theories
Because of this strong-coupling physics, one should calculate the IR zero in , I R, and resultant value of evaluated at I R to higher-lo op order (Ryttov and Shro ck, PRD 83, 056011 (2011), arXiv:1011.4542 and Pica and Sannino, PRD 83, 035013 (2011), arXiv:1011.5917; related work by Gardi, Grunberg, Karliner). Although co effs. in at 3 lo op order are scheme-dependent, results give a measure of accuracy of the 2-lo op calc. of the IR zero of , and similarly with evaluated at this IR zero. We make use of calculation of and up to 4-lo ops in M S scheme by Vermaseren, Larin, and van Ritbergen. The value of higher-lo op calculations has been amply shown in comparison of QCD predictions with experimental data, e.g., in M S scheme. Many contributions by authors originally and/or currently at INR, MSU, JINR: Chetyrkin, Gorishny, Kataev, Larin, Surguladze, Tkachov, Tarasov, Vladimirov, Zharkov...


3-lo op co efficient in function (in MS scheme) (Tarasov, Vladimirov, Zharkov; Larin and Vermaseren) 2857 54 205 1415 27

b3 =

C + Tf N

3 A

f 2

2C - 44 9

2 f

+(Tf Nf ) b3 < 0 for Nf I . Since from = 0 at two values:
3

Cf +

9 158 27

CA Cf - C
A

C

2 A

= -[2/(2 )](b1 + b2a + b3a2),

3

= 0 away

=

2 b
3

- b2 ±

b 2 - 4b 1 b 2

3

Since b2 < 0 and b3 < 0, can rewrite as = 2 |b 3 | - |b 2 | b 2 + 4b 1 |b 3 | 2
R,3

Soln. with - sqrt is negative, hence unphysical; soln. with + sqrt is I

.


We showed that with this b3 < 0, the value of the IR zero decreases when calculated at the 3-lo op level, i.e., I Pro of: I
R,2 R,3

<

I R,2

- I

R,3

=

2 2b 1 |b 3 | + b 2 - | b 2 | b 2 + 4b 1 |b 3 | = 2 2 |b 2 b 3 | The expression in square brackets is positive if and only if
2 (2b1|b3| + b2)2 - b2(b2 + 4b1|b3|) > 0 2 2

2 4 b 1 - - |b 2 | + |b 2 | |b 3 |

b 2 + 4b 1 |b 3 | 2

This difference is equal to the positive-definite quantity 4b2b2, which proves the 13 inequality.


In RS, Phys. Rev. D 87, 105005 (2013), arXiv:1301.3209 we have generalized this. If a scheme had b3 > 0 in I , then, since b2 0 at lower end of I , b2 - 4b1b3 < 0 2 in sqrt, so this scheme would not have a physical I R,3 in this region. Since the existence of the IR zero in at 2-lo op level is scheme-independent, one may require that a scheme should maintain this property to higher-lo op order, and hence that b3 < 0 for Nf I . So the inequality I
R,3

<

I R,2

holds in all such schemes, not just in MS.
4

The 4-lo op function is = -[2/(2 )](b1 + b2a + b3a2 + b4a3), so three zeros away from = 0; smallest (real positive) one as I R,4.

has

We give an analysis of the zeros of 4 in a general scheme in Phys. Rev. D 87, 105005 (2013). With MS, from 3- to 4-lo op level, slight increase: I R,4 > I small change, so overall, I R,4 < I R,2.

R,3

;

Our result of smaller fractional change in value of IR zero of at higher-lo op order agrees with expectation that calc. to higher lo op order should give more stable result.


Numerical values of I fermions in fund. rep.

R,n

at the n = 2, 3, 4 lo op level for SU(2), SU(3) and I R,2 11.42 2.83 1.26 0.595 0.231 2.21 1.23 0.754 0.468 0.278 0.143 0.0416 I R,3 1.645 1.05 0.688 0.418 0.196 0.764 0.578 0.435 0.317 0.215 0.123 0.0397 I R,4 2.395 1.21 0.760 0.444 0.200 0.815 0.626 0.470 0.337 0.224 0.126 0.0398

Nc Nf 26 27 28 29 2 10 3 10 3 11 3 12 3 13 3 14 3 15 3 16

(Perturbative calc. not applicable if I R,n to o large.) We have performed the corresponding higher-lo op calculations for SU(Nc) gauge theories with Nf fermions in the adjoint, symmetric and antisymmetric rank-2 tensor representations.


We prove a general result on the shift of an IR zero of when calculated at next higher order: assume fermion content is such that b2 < 0, so theory has a 2-lo op IR zero. Consider a scheme in which the b with = 3, ..., n + 1 have values that preserve the existence of the scheme-independent 2-lo op IR zero of at higher-lo op level (motivated physically). Use fact that theory is asymptotically free, so < 0 for 0 < < dn/d > 0 for I R,n. Expand
n IR

, and hence

in Taylor series around = I
n

R,n

:
R,n

=

I R,n

( - I

R,n

) + O ( - I

)2
R,(n+1)

Now calculate to the next-higher-lo op order, i.e., (n+1), and solve for I To determine whether I R,(n+1) is larger or smaller than I R,n , consider
(n+1)

.

-

n

= -2¯n+1n+ b

2


In a scheme where bn+1 > 0, this difference, evaluated at = I R,n, is negative, so, given that dn/d|I R,n > 0, to compensate for this, the zero shifts to the right, whereas if bn+1 < 0, the difference is positive, so the zero shifts to the left. If b >0, then I >

n+1

R,(n+1)

I R,n

If b

n+1

<0,

then I

R,(n+1)

<

I R,n

This general result is evident in our MS calculations.

b 3 < 0, b 4 > 0,

= =

I I

R,3

< >

I R,2

R,4

I R,3


It is of interest to calculate the anomalous dimension with series expansion =


m

for the fermion bilinear,

c a =



c ¯

=1

=1

where c = c/(4 ) is the -lo op co efficient. The one-lo op co eff. c1 = 6Cf is ¯ scheme-independent, the c with 2 are scheme-dependent and have been calculated up to 4-lo op level in M S scheme, as noted above. Denote calculated to n-lo op (n) level as n and, evaluated at the n-lo op value of the IR zero of , as I R,n n( = I R,n) In the IR chirally symmetric phase, an all-order calculation of evaluated at an all-order calculation of I R would be an exact property of the theory. In the bk. phase, just as the IR zero of is only an approx. IRFP, so also, the is ¯ only approx., describing the running of and the dynamically generated running fermion mass near the zero of having large-momentum behavior (k) (/k)2- . In both phases, is bounded above as < 2.


Illustrative numerical values of I R,n for SU(2) and SU(3) at the n = 2, 3, 4 lo op level and fermions in the fundamental representation: Nc Nf 27 28 29 2 10 3 10 3 11 3 12 3 13 3 14 3 15 3 16 I R,2 (2.67) 0.752 0.275 0.0910 (4.19) 1.61 0.773 0.404 0.212 0.0997 0.0272 I R,3 0.457 0.272 0.161 0.0738 0.647 0.439 0.312 0.220 0.146 0.0826 0.0258 I R,4 0.0325 0.204 0.157 0.0748 0.156 0.250 0.253 0.210 0.147 0.0836 0.0259

Plots of as fn. of Nf for SU(2) and SU(3):


Figure 1:



I R,

n-loop anomalous dimension I R,n at I 2 ; (ii) red: I R,3 ; (iii) brown: I R,4 .

R,n

for SU(2) with Nf fermions in fund. rep. (i) blue:


Figure 2:



I R,

n-loop anomalous dimension I R,n at I 2 ; (ii) red: I R,3 ; (iii) brown: I R,4 .

R,n

for SU(3) with Nf fermions in fund. rep: (i) blue:


A necessary condition for a perturbative calculation to be reliable is that higher-order contributions do not mo dify the result to o much. We find that the 3-lo op and 4-lo op results are closer to each other for a larger range of Nf than the 2-lo op and 3-lo op results. We have also done higher-lo op calcs. for a supersymmetric gauge theory in Ryttov and Shro ck, Phys. Rev. D 85, 076009 (2012) (arXiv:1202.1297) - not discussed here. So our higher-lo op calcs. of I R and allow us to probe the theory reliably down to smaller values of Nf and thus stronger couplings, closer to Nf ,cr . Of course, perturbative calculations are not applicable when is to o large. We have also performed these higher-lo op calculations for larger fermion reps. R. In general, we find that, for a given Nc, R, and Nf , the values of I R,n calculated to 3-lo op and 4-lo op order are smaller than the 2-lo op value.


Example of a Comparison with Lattice Measurements
For SU(3) with Nf = 12, we calculate
I R,2

= 0.77,



I R,3

= 0.31,



I R,4

= 0.2 5

some lattice results (N.B.: error estimates do not include all systematic uncertainties) = 0.414 ± 0.016 (Appelquist et al., PRD 84, 054501 (2011), IR -sym.) 0.35 (DeGrand, PRD 84, 116901 (2011), IR -sym.) 0.2 < < 0.4 (Kuti et al. (metho d-dep.) arXiv:1205.1878, arXiv:1211.3548, 1211.6164, PTP, finding SSB) = 0.4 - 0.5 (Y. Aoki et al., (LatKMI) PRD 86, 054506 (2012))

= 0.27(3) (Hasenfratz et al., arXiv:1207.7162; = 0.32(3), arXiv:1301.1355, IR -sym.) So here the 2-lo op value is larger than, and the 3-lo op and 4-lo op values closer to, these lattice measurements. Thus, our higher-lo op calculations of yield better agreement with these lattice measurements than two-lo op calculations.


Further Higher-Loop Structural Properties of
In addition to I n include
R,n

, further interesting structural properties of the n-lo op beta fn.

· the derivative

I R,n

· the magnitude and lo cation of the minimum in In quasi-scale-invariant case where I R EWSB mo dels depends on how small order, on I R,n, via the series expansio n() =
I R,n



dn d

evaluated at I

R,n

.

n

> cr , dilaton mass relevant in dynamical is for near to I R and hence, at n-lo op n of n around I R,n, ) + O ( - I )2

( - I

R,n

R,n

We have calculated these structural properties analytically and numerically (RS, PRD 87, 105005 (2013), arXiv:1301.3209.


Derivative of 2-lo op function at I

R,2

:



I R,2

2(11CA - 4Tf Nf )2 2b 2 2b 2 1 1 = = =- 2 b2 |b 2 | 3[4(5CA + 3Cf )Tf Nf - 34CA] - 4|b2|(b2 + b1|b3|) + (b2 + 2b1|b3|) 2 2 b 2 + 4b 1 |b 3 | 2

At 3-lo op level:
I R,3

=

1 |b 3 |
2

We prove a general inequality: for a given gauge group G, fermion rep. R, and Nf I (in a scheme with b3 < 0, which thus preserves the existence of the 2-lo op IR zero in at 3-lo op level),
I R,3 I R,2



<

We carry out a similar analysis of the derivative of the 4-lo op function evaluated at I R,4, denoted I R,4, and find a similar decrease from 3-lo op to 4-lo op order. Some numerical values:


Nc Nf 27 28 29 2 10 3 10 3 11 3 12 3 13 3 14 3 15 3 16

I R,2 1.20 0.400 0.126 0.0245 1.52 0.720 0.360 0.174 0.0737 0.0227 0.00221

I R,3 0.728 0.318 0.115 0.0239 0.872 0.517 0.2955 0.156 0.0699 0.0223 0.00220

I R,4 0.677 0.300 0.110 0.0235 0.853 0.498 0.282 0.149 0.0678 0.0220 0.00220

Illustrative figures for SU(2) with Nf = 8 fermions and SU(3) with Nf = 12 fermions:


0.15

0.1


0.05

0

0.2

0.4

0.6

0.8

1

1.2

1.4


­0.05

­0.1

Figure 3:



n

for SU(2), Nf = 8, at n = 2, 3, 4 loops. From bottom to top, curves are 2, 4, 3.


0.1 0.08 0.06


0.04 0.02 0 ­0.02 ­0.04 ­0.06 ­0.08 ­0.1

0.2

0.4

0.6

0.8

1



Figure 4:



n

for SU(3), Nf = 12, at n = 2, 3, 4 loops. From bottom to top, curves are 2, 4, 3.


Interesting property: for R = fund. rep., I R,n Nc, I R,n, and other structural properties of n are similar in theories with different values of Nc and Nf if they have equal or similar values of r = Nf /Nc. This motivates a study of the UV to IR evolution of an SU(Nc) gauge theory with N fermions in the fundamental rep. in the 't Ho oft-Veneziano limit Nc , Nf with Nf r Nc fixed, (µ)Nc (µ) indep. of N
c f

Denote this as the LNN (large Nc, large Nf ) limit. Asymptotic freedom requires r < 11/2. ,2 has IR zero for 34/13 < r < 11/2, i.e., 2.62 < r < 5.5. We have carried out this study in RS, PRD 87, 116007 (2013), arXiv:1302.5434. Our results provide a unified quantitative understanding of the similarities in UV to IR evolution of SU(Nc) theories with different Nc and Nf but similar r . With = Nc and x = aNc = /(4 ), define a rescaled beta function that is finite in the LNN limit: d dt = lim N
LN N c


Denote the IR zero of n-lo op as
I R,4

I R,n

. By same type of analysis as before, we find
I R,2

I R,3



< >

I R,3

if 2.615 < r < 3.119 if 3.119 < r < 5.500

I R,4

I R,3

Numerical values given in next table. The magnitude of the fractional difference | is reasonably small.
I R,4



-

I R,3

|

I R,4


r 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4

I R,2 28.274 12.566 7.606 5.174 3.731 2.774 2.095 1.586 1.192 0.8767 0.6195 0.4054 0.2244 0.06943

I R,3 3.573 2.938 2.458 2.076 1.759 1.489 1.252 1.041 0.8490 0.6725 0.5083 0.3538 0.2074 0.06769

I R,4 3.323 2.868 2.494 2.168 1.873 1.601 1.349 1.115 0.9003 0.7038 0.5244 0.3603 0.2089 0.06775


We also study the anomalous dimension

m

¯ of in this LNN limit.
I R,n

Denote the n-lo op evaluated at n-lo op IR zero of as e.g., at 2-lo op level,

.



I R,2

=

(11 - 2r )(1009 - 158r + 40r 2) 12(13r - 34)2

with similar results for higher-lo op order.


Numerical values: r 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 General inequalities as before: I R,2 1.853 1.178 0.7847 0.5366 0.3707 0.2543 0.1696 0.1057 0.05620 0.01682 I R,3 0.5201 0.4197 0.3414 0.2771 0.2221 0.1735 0.1294 0.08886 0.05123 0.01637 , I R,4 0.3083 0.3061 0.2877 0.2664 0.2173 0.1745 0.1313 0.08999 0.05156 0.01638 <

I R,3

<

I R,2

I R,4

I R,2


We have studied the approach to the LNN limit and find that this is quite rapid, with 2 leading correction terms suppressed by 1/Nc . For example, I
R,2

Nc =

4 (11 - 2r ) 13r - 34

+

(34 - 13r )2N

12 r (11 - 2r )
2 c

+O

1 N
4 c

+

12(13r - 34)2 (11 - 2r )(18836 - 5331r + 648r 2 - 140r 3) (13r - 34)3N
2 c



I R,2

=

(11 - 2r )(1009 - 158r + 40r 2) +O 1 N
4 c

This explains the approximate universality that is exhibited in calculations of these quantities for different (finite) values of Nc and Nf with similar or identical values of r .


Study of Scheme Dependence in Calculation of IR Fixed Point
Since co effs. bn in n, and hence also I R,n , are scheme-dependent for n 3, it is important to assess the effects of this scheme dependence. Extensive studies of scheme dependence in QCD, relevant for high-energy quark-parton pro cesses where s(µ) is small, governed by the UV fixed point (Bro dsky, Lepage, MacKenzie; Celmaster and Gonsalves; Stevenson; Garkusha, Gorishny, Kataev, Larin, Surguladze; Gracey; Bro dsky, Mojaza, Wu...) Here we fo cus not on theories such as QCD near the UV fixed point but on scheme dependence in calculation of an approx. or exact infrared fixed point of an asymptotically free theory: results in Ryttov and Shro ck, PRD 86, 065032 (2012), arXiv:1206.2366; PRD 86, 085005 (2012), arXiv:1206.6895; and Shro ck, PRD 88, 036003 (2013), arXiv:1305.6524.


A scheme transformation (ST) is a map between and or equivalently, a and a, where a = /(4 ) of the form a = af (a) with f (0) = 1 to keep UV properties unchanged. Write
smax smax

f (a) = 1 +
s =1

ks(a)s = 1 +
s =1

¯ ks()s ,

¯ where ks = ks/(4 )s, and smax may be finite or infinite. The Jacobian J = da/da = d/d = 1 + J = 1 a t a = a = 0 .
smax s =1

(s + 1)ks(a)s, satisfying

After scheme transformation is applied, beta function in new scheme is d d d = = J -1 . dt d dt = -2
=1





b (a) = -2

=1

¯ () , b

where ¯ = b /(4 ). b


We calculate the b as functions of the b and ks. At 1-lo op and 2-lo op, this yields the well-known results b = b1 , b = b2 1 2 We find
2 b = b3 + k1b2 + (k1 - k2)b1 , 3

2 3 b = b4 + 2k1b3 + k1 b2 + (-2k1 + 4k1k2 - 2k3)b 4

1

2 3 b = b5 + 3k1b4 + (2k1 + k2)b3 + (-k1 + 3k1k2 - k3)b 5 4 2 2 +(4k1 - 11k1 k2 + 6k1k3 + 4k2 - 3k4)b

2

1

etc. at higher-lo op order.


A physically acceptable ST must satisfy several conditions: C1: the ST must map a (real positive) to a real positive , since a map taking > 0 to = 0 would be singular, and a map taking > 0 to a negative or complex would violate the unitarity of the theory. C2: the ST should not map a mo derate value of , where perturbation theory is applicable, to a value of so large that pert. theory is inapplicable. C3: J should not vanish, or else there would be a pole in


C4: Existence of an IR zero of is a scheme-independent property, so the ST should satisfy the condition that has an IR zero if and only if has an IR zero. These conditions can always be satisfied by an ST near the UVFP at = = 0, but they are not automatic, and can be quite restrictive at an IRFP.


For example, consider the ST (dependent on a parameter r ) tanh(r a) a= r with inverse 1 1 + ra a= ln 2r 1 - ra


This is acceptable for small a, but if a > 1/r , i.e., > 4 /r , it maps a real to a complex and hence is physically unacceptable. For, say, r = 8 , this pathology can o ccur at the mo derate value = 0.5. We have constructed several STs that are acceptable at an IRFP and have studied scheme dependence of the IR zero of n using these. For example, a= with inverse a=


sinh(r a) r 1 + (r a)2

1 r

ln r a +

We find reasonably small scheme-dependence for mo derate I R.


Since the bn with n 3 are scheme-dependent, one might possible, at least in the vicinity of the UVFP at = = transformations that would set b = 0 for some range of n n that would do this for all n 3, so that would consist 2-lo op terms ('t Ho oft scheme).

expect that it would be 0, to construct a scheme 3, and, indeed a ST only of the 1-lo op and

We have constructed an explicit scheme transformation that do es this in the vicinity of the UVFP at = = 0. To construct this ST, first, solve eq. b = 0 for k2, obtaining 3 k2 = b b
3 1

+

b b

2 1

2 k1 + k1

Next, substitute this into expression for b and solve eq. b = 0, obtaining 4 4 3b 3 5b 2 2 b4 3 + k1 + k1 + k1 k3 = 2b 1 b1 2b 1 Continue this pro cedure iteratively. Our studies give a quantitative assessment of the scheme dep endence of an IR zero of at lo op order n 3.


Conclusions
· Neutrino masses and lepton mixing are of great significance; neutrino electromagnetic properties can also be important. · Understanding the UV to IR evolution of an asymptotically free gauge theory and the nature of the IR behavior is of fundamental interest and can be relevant to exploring BSM physics. · Our higher-lo op calculations give information on this UV to IR flow and on determination of I R,n and I R,n, provide comparison with lattice measurements. · Results on the limit Nc , Nf with Nf /Nc fixed yield understanding of similarities in UV to IR flows in theories with different Nc and Nf but similar r . · We have investigated effects of scheme-dependence of IR zero in higher-lo op calculations and have pointed out that scheme transformations are subject to conditions that are easily satisfied at a UVFP but are a significant constraint at an IRFP.